A Schur-Nevanlinna type algorithm for the truncated matricial Hausdorff moment problem
aa r X i v : . [ m a t h . C A ] M a y A Schur–Nevanlinna type algorithm for thetruncated matricial Hausdorff moment problem
Bernd Fritzsche Bernd Kirstein Conrad Mädler
The main goal of this paper is to achieve a parametrization of the solution setof the truncated matricial Hausdorff moment problem in the non-degenerate anddegenerate situation. We treat the even and the odd cases simultaneously. Our ap-proach is based on Schur analysis methods. More precisely, we use two interrelatedversions of Schur-type algorithms, namely an algebraic one and a function-theoreticone. The algebraic version, worked out in our former paper [29], is an algorithmwhich is applied to finite or infinite sequences of complex matrices. The construc-tion and discussion of the function-theoretic version is a central theme of this paper.This leads us to a complete description via Stieltjes transform of the solution setof the moment problem under consideration. Furthermore, we discuss special solu-tions in detail.
Mathematics Subject Classification (2010)
Keywords:
Truncated matricial Hausdorff moment problem, Schur–Nevanlinna type algorithm,parametrization of the solution set via Stieltjes transform.
1. Introduction
The main aim of this paper is to work out an algorithm of Schur–Nevanlinna type for func-tions, which leads via Stieltjes transforms to a full description of the solution set of the ma-tricial Hausdorff moment problem in the general case. In order to realize this goal, we useour former investigations in [27–29] on matricial Hausdorff moment sequences, where such se-quences were studied from the point of view of Schur analysis and an algebraic version of acorresponding Schur–Nevanlinna type algorithm was worked out. The synthesis of these twoSchur–Nevanlinna type algorithms will provide us finally the desired result.This strategy was already used by the authors to study the matricial Hamburger momentproblem (see [23, 32]) and the matricial Stieltjes moment problem (see [24, 25, 31]).It will turn out that the case of the matricial Hausdorff moment problem is much moredifficult in comparison with the matricial problems named after Hamburger and Stieltjes. Thisphenomenon could be already observed in the discussion of the so-called non-degenerate case,where the Hamburger problem (see [14, 41]) and the Stieltjes problem (see [15, 16, 18, 19])were studied considerably earlier than the Hausdorff problem (see [10, 11]). The main reasonfor the greater complexity of the Hausdorff moment problem is caused by the fact that thelocalization of the measure in a prescribed compact interval of the real axis requires to satisfy1imultaneously more conditions. This implies that the possible moment sequences have a morecomplicated structure (see [27–29]).Continuing the work done in [5, 10, 11] A. E. Choque-Rivero [6–9] investigated further as-pects of the non-degenerate truncated matricial Hausdorff moment problem. As in [5, 10, 11]he distinguished between the case of an odd or even number of prescribed matricial moments.The approach used in [5, 10, 11] is based on V. P. Potapov’s method of Fundamental MatrixInequalities. In the scalar case q = 1 the classical method used in [43] or [42, Ch. 4, § 7]is based on the application of orthogonal polynomials. A. E. Choque-Rivero [6] obtained amatrix generalization of the Krein–Nudelman representation of the resolvent matrix by usingfour families of orthogonal matrix polynomials on the interval [ α, β ]. From the point of viewof V. P. Potapov the Schur algorithm is interpreted as a multiplicative decomposition of a J -contractive matrix function into simplest elementary factors. Such multiplicative decom-positions were constructed for the resolvent matrix for the moment problem under study byA. E. Choque-Rivero in [7, 8].An important feature of this paper is to achieve a simultaneous treatment of the even andodd truncated matricial Hausdorff moment problems in the general case. Our strategy is basedon the application of Schur analysis methods.This paper is organized as follows. In Section 2, we state some general facts on matricialpower moment problems on Borel subsets of the real axis. In Section 3, we summarize essentialinsights about the structure of matricial [ α, β ]-Hausdorff moment sequences, which were mostlyobtained in our former papers [17–19]. A key observation is that we do not treat the originalmatricial moment problem, but an equivalent problem in the class R q ( C \ [ α, β ]) of holomor-phic matrix functions. In Section 4, we summarize some facts on the class R q ( C \ [ α, β ]), whichare needed in the sequel. In Section 5, we formulate a problem for functions of R q ( C \ [ α, β ])which is equivalent to the original matricial moment problem. This equivalence is causedby [ α, β ]-Stieltjes transform. In [28], we constructed a Schur–Nevanlinna type algorithm fornon-negative Hermitian q × q measures on [ α, β ] by translating the Schur–Nevanlinna type al-gorithm for matricial [ α, β ]-Hausdorff moment sequences into the language of measures. In Sec-tion 6, we translate now this algorithm via [ α, β ]-Stieltjes transform into the class R q ( C \ [ α, β ]).On this way our main goal is to achieve a description of all solutions of the truncated matri-cial [ α, β ]-Hausdorff moment problem via a linear fractional transformation of matrices. Thisrequires to find the generating matrix function of this transformation and the correspondingdomain. In the first step, we concentrate on that domain. Remember that in [10] we al-ready studied the problem under consideration in the non-degenerate case, however by use ofPotapov’s method of fundamental matrix inequalities. Doing this, we were led to a particularclass PR q ( C \ [ α, β ]) of ordered pairs of q × q matrix-valued functions which are meromorphicin C \ [ α, β ] (see [10, Def. 5.2]). In Section 7, we summarize some important facts about the class PR q ( C \ [ α, β ]) and present an example of a remarkable element of this class (see Example 7.15).The experiences from [10, 11] teach us that it is necessary to introduce an equivalence relationwithin PR q ( C \ [ α, β ]). In order to take into account possible degeneracies of the moment prob-lem, we have to single out an appropriate subclass of PR q ( C \ [ α, β ]), which is adapted to theprescribed matricial moments. Furthermore, we have to ensure that the construction of thissubclass stands in harmony with the above mentioned equivalence relation in PR q ( C \ [ α, β ]).The just mentioned two themes are treated in Section 8. The goal of the following considera-tions is to prepare basic instruments for the version of our Schur–Nevanlinna type algorithmfor functions. This algorithm should stand in correspondence with the Schur–Nevanlinna typealgorithm for matricial [ α, β ]-non-negative definite sequences, which was developed in [29].2 remarkable feature of this version is that this algorithm contains two elements of differ-ent nature. More precisely, the first step of the algorithm differs from the remaining steps.There occur (equivalence classes) of ordered pairs of matrix-valued functions in the first step,whereas the further steps require only matrix-valued functions. In Sections 9 and 10, we studythe corresponding transformations and its inverses for the first and remaining steps, resp. Thetransformations are defined by using Moore–Penrose inverses of matrices. It will turn out thatunder special conditions, which will be indeed satisfied in the case of interest for us, thesetransformations can be rewritten as usual linear fractional transformations of matrices the gen-erating matrix-valued functions of which are quadratic 2 q × q matrix polynomials. Having acloser look at the considerations in Sections 9 and 10 one can observe that the basic tools usedthere are of rather algebraic nature. In Sections 11 and 12, we study the elementary steps ofthe forward and backward algorithm in more detail, resp. Namely, we turn our attention tothe concrete classes of meromorphic matrix-valued functions occurring there. Moreover, wedemonstrate that these elementary steps of the algorithm for functions are concordant withthe elementary steps of the algebraic algorithm applied to the matricial moment sequences. InSection 13, we check (see Theorem 13.6) that the iteration of the elementary steps leads to aparametrization of the set of [ α, β ]-Stieltjes transforms of all solutions of the original matri-cial moment problem. In Section 14, we translate Theorem 13.6 into the language of linearfractional transformations of matrices and obtain our main results Theorems 14.2 and 14.5.In Section 15, via [ α, β ]-Stieltjes transform we determine all those solutions σ of the momentproblem associated with a sequence ( s j ) mj =0 for which the sequence ( s j ) m +1 j =0 , where s m +1 is the( m + 1)-th power moment of σ , is [ α, β ]-completely degenerate.The main object of study in Section 16 is that solution of the moment problem associatedwith a sequence ( s j ) mj =0 which corresponds to the central extension of ( s j ) mj =0 . We determinethe position of the [ α, β ]-Stieltjes transform of this solution within the general parametrizationobtained in Theorem 14.2.In several appendices we summarize some needed facts on various topics such as particularaspects of matrix theory, non-negative Hermitian measures and corresponding integration the-ory, Stieltjes of non-negative Hermitian measures on the real line, ordered pairs of matricescorresponding to linear relations, linear fractional transformations of matrices, holomorphicmatrix-valued functions.
2. Matricial moment problems on Borel subsets of the real axis
In this section, we are going to formulate a certain class of matricial power moment problems.Before doing this, we have to introduce some terminology. We denote by Z the set of allintegers. Let N := { n ∈ Z : n ≥ } . Furthermore, we write R for the set of all real numbersand C for the set of all complex numbers. In the whole paper, p and q are arbitrarily fixedintegers from N . We write C p × q for the set of all complex p × q matrices and C p is short for C p × . When using m, n, r, s, . . . instead of p, q in this context, we always assume that theseare integers from N . We write A ∗ for the conjugate transpose of a complex p × q matrix A .Denote by C q × q < (resp. C q × q ≻ ) the set of all complex non-negative (resp. positive) Hermitian q × q matrices. If X is a subset of C q × r and if A ∈ C p × q , then let A X := { AX : X ∈ X } .Let ( X , X ) be a measurable space. Each countably additive mapping whose domain is X andwhose values belong to C q × q < is called a non-negative Hermitian q × q measure on ( X , X ). Forthe integration theory with respect to non-negative Hermitian measures, we refer to Kats [38]3nd Rosenberg [47]. For the convenience of the reader, a summary concerning this matter,sufficient for our purposes, is given in Appendix B.Let B R (resp. B C ) be the σ -algebra of all Borel subsets of R (resp. C ). In the whole paper,Ω stands for a non-empty set belonging to B R . Let B Ω be the σ -algebra of all Borel subsets ofΩ and let M < q (Ω) be the set of all non-negative Hermitian q × q measures on (Ω , B Ω ). Observethat M < (Ω) coincides with the set of ordinary measures on (Ω , B Ω ) with values in [0 , ∞ ).Let N := { m ∈ Z : m ≥ } . Throughout this paper, κ is either an integer from N or ∞ . Inthe latter case, we have 2 κ := ∞ . Given υ, ω ∈ R ∪ {−∞ , ∞} , we set Z υ,ω := { k ∈ Z : υ ≤ k ≤ ω } . Let M < q,κ (Ω) be the set of all µ ∈ M < q (Ω) such that for each j ∈ Z ,κ the power function x x j defined on Ω is integrable with respect to µ . If µ ∈ M < q,κ (Ω), then, for all j ∈ Z ,κ , thematrix s ( µ ) j := Z Ω x j µ (d x ) (2.1)is called (power) moment of µ of order j . Obviously, we have M < q (Ω) = M < q, (Ω) ⊆ M < q,ℓ (Ω) ⊆M < q,ℓ +1 (Ω) ⊆ M < q, ∞ (Ω) for every choice of ℓ ∈ N and, furthermore, s ( µ )0 = µ (Ω) for all µ ∈ M < q (Ω). If Ω is bounded, then one can easily see that M < q (Ω) = M < q, ∞ (Ω). We now statethe general form of the moment problem lying in the background of our considerations: Problem MP [Ω; ( s j ) κj =0 , =] : Given a sequence ( s j ) κj =0 of complex q × q matrices, parametrizethe set M < q,κ [Ω; ( s j ) κj =0 , =] of all σ ∈ M < q,κ (Ω) satisfying s ( σ ) j = s j for all j ∈ Z ,κ .In the whole paper, let α and β be two arbitrarily given real numbers satisfying α < β andlet δ := β − α . In what follows, we mainly consider the case that Ω is the compact interval[ α, β ] of the real axis R . As mentioned above, we have M < q ([ α, β ]) = M < q, ∞ ([ α, β ]).Since each solution of MP [[ α, β ]; ( s j ) κj =0 , =] generates in a natural way solutionsof MP [[ α, ∞ ); ( s j ) κj =0 , =], MP [( −∞ , β ]; ( s j ) κj =0 , =], and MP [ R ; ( s j ) κj =0 , =], we will also use re-sults concerning the treatment of these moment problems.
3. Matricial [ α, β ] -Hausdorff moment sequences In this section, we recall a collection of results on the matricial Hausdorff moment problemand corresponding moment sequences of non-negative Hermitian q × q measures on the interval[ α, β ], which are mostly taken from [27–29]. To state a solvability criterion, we introduce therelevant class of sequences of complex matrices. Notation . Let ( s j ) κj =0 be a sequence of complex p × q matrices. Then let the block Hankelmatrices H n , K n , and G n be given by H n := [ s j + k ] nj,k =0 for all n ∈ N with 2 n ≤ κ , by K n := [ s j + k +1 ] nj,k =0 for all n ∈ N with 2 n + 1 ≤ κ , and by G n := [ s j + k +2 ] nj,k =0 for all n ∈ N with 2 n + 2 ≤ κ , resp.To emphasize that a certain (block) matrix X is built from a sequence ( s j ) κj =0 , we sometimeswrite X h s i for X . Notation . Assume κ ≥ s j ) κj =0 be a sequence of complex p × q matrices. Thenlet the sequences ( a j ) κ − j =0 and ( b j ) κ − j =0 be given by a j := − αs j + s j +1 and b j := βs j − s j +1 , κ ≥
2, then let the sequence ( c j ) κ − j =0 be given by c j := − αβs j + ( α + β ) s j +1 − s j +2 . Similarly to our more algebraic considerations [27–29] it will turn out that a characteristicfeature of MP [[ α, β ]; ( s j ) κj =0 , =] is to analyze and organize the interplay between the four matrixsequences ( s j ) κj =0 , ( a j ) κ − j =0 , ( b j ) κ − j =0 and ( c j ) κ − j =0 under the view of corresponding block Hankelmatrices generated by them. For each matrix X k = X h s i k built from the sequence ( s j ) κj =0 ,we denote (if possible) by X α,k, • := X h a i k , by X • ,k,β := X h b i k , and by X α,k,β := X h c i k thecorresponding matrix built from the sequences ( a j ) κ − j =0 , ( b j ) κ − j =0 , and ( c j ) κ − j =0 instead of ( s j ) κj =0 ,resp.In view of Notation 3.1, we get in particular H α,n, • = − αH n + K n and H • ,n,β = βH n − K n for all n ∈ N with 2 n + 1 ≤ κ and H α,n,β = − αβH n + ( α + β ) K n − G n for all n ∈ N with 2 n + 2 ≤ κ . In the classical case α = 0 and β = 1, we have furthermore a j = s j +1 and b j = s j − s j +1 for all j ∈ Z ,κ − and c j = s j +1 − s j +2 for all j ∈ Z ,κ − . Remark . Let ( s j ) κj =0 be a sequence of complex p × q matrices. Then δs j = a j + b j and δs j +1 = βa j + αb j for all j ∈ Z ,κ − . Furthermore, δs j +2 = β a j + α b j − δc j for all j ∈ Z ,κ − . Definition 3.4.
Let F < q, ,α,β (resp. F ≻ q, ,α,β ) be the set of all sequences ( s j ) j =0 of complex q × q matrices, for which the block Hankel matrix H is non-negative (resp. positive) Hermitian,i. e., for which s ∈ C q × q < (resp. s ∈ C q × q ≻ ) holds true. For each n ∈ N , denote by F < q, n,α,β (resp. F ≻ q, n,α,β ) the set of all sequences ( s j ) nj =0 of complex q × q matrices, for which the blockHankel matrices H n and H α,n − ,β are both non-negative (resp. positive) Hermitian. For each n ∈ N , denote by F < q, n +1 ,α,β (resp. F ≻ q, n +1 ,α,β ) the set of all sequences ( s j ) n +1 j =0 of complex q × q matrices for which the block Hankel matrices H α,n, • and H • ,n,β are both non-negative(resp. positive) Hermitian. Furthermore, denote by F < q, ∞ ,α,β (resp. F ≻ q, ∞ ,α,β ) the set of allsequences ( s j ) ∞ j =0 of complex q × q matrices satisfying ( s j ) mj =0 ∈ F < q,m,α,β (resp. ( s j ) mj =0 ∈F ≻ q,m,α,β ) for all m ∈ N . The sequences belonging to F < q, ,α,β , F < q, n,α,β , F < q, n +1 ,α,β , or F < q, ∞ ,α,β (resp. F ≻ q, ,α,β , F ≻ q, n,α,β , F ≻ q, n +1 ,α,β , or F ≻ q, ∞ ,α,β ) are said to be [ α, β ] -non-negative definite (resp. [ α, β ] -positive definite ).(Note that in [27], the sequences belonging to F < q,κ,α,β were called [ α, β ] -Hausdorffnon-negative definite .) A necessary and sufficient condition for the solvability of MP [[ α, β ]; ( s j ) κj =0 , =] is the following: Theorem 3.5 (cf. [10, Thm. 1.3] and [11, Thm. 1.3]) . Let ( s j ) κj =0 be a sequence of complex q × q matrices. Then M < q,κ [[ α, β ]; ( s j ) κj =0 , =] = ∅ if and only if ( s j ) κj =0 ∈ F < q,κ,α,β . Since Ω = [ α, β ] is bounded, one can easily see that M < q ([ α, β ]) = M < q, ∞ ([ α, β ]), i. e., thepower moment s ( σ ) j defined by (2.1) exists for all j ∈ N . If σ ∈ M < q ([ α, β ]), then we call( s ( σ ) j ) ∞ j =0 given by (2.1) the sequence of power moments associated with σ .Given the complete sequence of prescribed power moments ( s j ) ∞ j =0 , the moment problemon the compact interval Ω = [ α, β ] differs from the moment problems on the unbounded setsΩ = [ α, ∞ ) and Ω = R in having necessarily a unique solution, assumed that a solution exists:5 roposition 3.6. If ( s j ) ∞ j =0 ∈ F < q, ∞ ,α,β , then the set M < q, ∞ [[ α, β ]; ( s j ) ∞ j =0 , =] consists of ex-actly one element. Proposition 3.6 is a well-known result, which can be proved, in view of Theorem 3.5, usingthe corresponding result in the scalar case q = 1 (see, [34] or [1, Thm. 2.6.4]).We can summarize Proposition 3.6 and Theorem 3.5 for κ = ∞ : Proposition 3.7.
The mapping Ξ [ α,β ] : M < q ([ α, β ]) → F < q, ∞ ,α,β given by σ ( s ( σ ) j ) ∞ j =0 is welldefined and bijective. For each n ∈ N , denote by H < q, n the set of all sequences ( s j ) nj =0 of complex q × q matrices,for which the corresponding block Hankel matrix H n is non-negative Hermitian. Furthermore,for each ι ∈ N ∪ {∞} and each non-empty set X , denote by S ι ( X ) the set of all sequences( X j ) ιj =0 from X . Obviously, F < q, ,α,β coincides with the set of all sequences ( s j ) j =0 with s ∈ C q × q < . Furthermore, we have F < q, n,α,β = n ( s j ) nj =0 ∈ H < q, n : ( c j ) n − j =0 ∈ H < q, n − o (3.1)for all n ∈ N and F < q, n +1 ,α,β = n ( s j ) n +1 j =0 ∈ S n +1 ( C q × q ) : n ( a j ) nj =0 , ( b j ) nj =0 o ⊆ H < q, n o (3.2)for all n ∈ N . Note that the following Propositions 3.8 and 3.9, which are proved in a purelyalgebraic way in [27], can also be obtained immediately from Theorem 3.5. Proposition 3.8 (cf. [27, Prop. 7.7(a)]) . If ( s j ) κj =0 ∈ F < q,κ,α,β , then ( s j ) mj =0 ∈ F < q,m,α,β forall m ∈ Z ,κ . In view of Proposition 3.8, the definition of the class F < q, ∞ ,α,β seems to be natural. Proposition 3.9 ( [27, Prop. 9.1]) . Let ( s j ) κj =0 ∈ F < q,κ,α,β . If κ ≥ , then { ( a j ) κ − j =0 , ( b j ) κ − j =0 } ⊆F < q,κ − ,α,β . If κ ≥ , then furthermore ( c j ) κ − j =0 ∈ F < q,κ − ,α,β . We write R ( A ) := { Ax : x ∈ C q } and N ( A ) := { x ∈ C q : Ax = O p × } for the column spaceand the null space of a complex p × q matrix A , resp. Denote by D p × q,κ the set of all sequences( s j ) κj =0 of complex p × q matrices satisfying S κj =0 R ( s j ) ⊆ R ( s ) and N ( s ) ⊆ T κj =0 N ( s j ). Remark . Let ( s j ) κj =0 ∈ F < q,κ,α,β . Then ( s j ) κj =0 ∈ D q × q,κ . If κ ≥ { ( a j ) κ − j =0 , ( b j ) κ − j =0 } ⊆ D q × q,κ − . If κ ≥
2, moreover ( c j ) κ − j =0 ∈ D q × q,κ − .The set C q × q H := { M ∈ C q × q : M ∗ = M } of Hermitian matrices from C q × q is a partiallyordered R -vector space with positive cone C q × q < . For two complex q × q matrices A and B , wewrite A B or B < A if A, B ∈ C q × q H and B − A ∈ C q × q < are fulfilled. This partial order onthe set of Hermitian matrices is sometimes called Löwner semi-ordering . Lemma 3.11 ( [28, Lem. 5.7]) . Let ( s j ) κj =0 ∈ F < q,κ,α,β . Then s j ∈ C q × q H for all j ∈ Z ,κ and s k ∈ C q × q < for all k ∈ N with k ≤ κ . Furthermore, αs k s k +1 βs k for all k ∈ N with k + 1 ≤ κ . Let O p × q be the zero matrix from C p × q and let I q := [ δ jk ] qj,k =1 be the identity matrix from C q × q , where δ jk is the Kronecker delta. Sometimes, if the size is clear from the context, wewill omit the indices and write O and I , resp. Taking into account Remark A.27, we obtainfrom Lemma 3.11: 6 emark . If ( s j ) κj =0 ∈ F < q,κ,α,β , then R ( a k ) ∪ R ( b k ) ⊆ R ( s k ) and N ( s k ) ⊆ N ( a k ) ∩N ( b k ) for all k ∈ N with 2 k ≤ κ − F < q,m,α,β can always be extended to sequences from F < q,ℓ,α,β for all ℓ ∈ Z m +1 , ∞ , which is due to the fact that a non-negative Hermitian measure on the boundedset [ α, β ] possesses power moments of all non-negative orders. One of the main results in [27]states that the possible one-step extensions s m +1 ∈ C q × q of a sequence ( s j ) mj =0 to an [ α, β ]-non-negative definite sequence ( s j ) m +1 j =0 fill out a matricial interval. In order to give an exactdescription of this interval, we are now going to introduce several matrices and recall theirrole in the corresponding extension problem for [ α, β ]-non-negative definite sequences, studiedin [27].Given an arbitrary n ∈ N and arbitrary rectangular complex matrices A , A , . . . , A n , wewrite col ( A j ) nj =1 = col( A , A , . . . , A n ) (resp., row ( A j ) nj =1 := [ A , A , . . . , A n ]) for the blockcolumn (resp., block row) built from the matrices A , A , . . . , A n if their numbers of columns(resp., rows) are all equal. Notation . Let ( s j ) κj =0 be a sequence of complex p × q matrices. For all ℓ, m ∈ N with ℓ ≤ m ≤ κ , then let y ℓ,m := col ( s j ) mj = ℓ and z ℓ,m := row ( s j ) mj = ℓ .The block Hankel matrix H n admits the following block representations: Remark . If κ ≥ s j ) κj =0 is a sequence of complex p × q matrices, then H n = (cid:2) H n − y n, n − z n, n − s n (cid:3) and H n = (cid:2) s z ,n y ,n G n − (cid:3) for all n ∈ N with 2 n ≤ κ .In this paper, the Moore–Penrose inverse of a complex matrix plays an important role. Foreach matrix A ∈ C p × q , there exists a unique matrix X ∈ C q × p , satisfying the four equations AXA = A, XAX = X, ( AX ) ∗ = AX, and ( XA ) ∗ = XA (3.3)(see e. g. [13, Prop. 1.1.1]). This matrix X is called the Moore–Penrose inverse of A and isdenoted by A † . Concerning the concept of Moore–Penrose inverse we refer to [46], [4, Ch. 1],and [2, Ch. 1]. For our purposes, it is convenient to apply [13, Sec. 1.1].If (cid:2) A BC D (cid:3) is the block representation of a complex ( p + q ) × ( r + s ) matrix M with p × r block A , then the matrix M/A := D − CA † B (3.4)is called the Schur complement of A in M . Concerning a variety of applications of this conceptin a lot of areas of mathematics, we refer to [54].In the paper, various kinds of concrete Schur complements in block matrices will play anessential role. By virtue of Remark 3.14, we use in the sequel the following notation: Notation . If ( s j ) κj =0 is a sequence of complex p × q matrices, then let L := H and let L n := H n /H n − for all n ∈ N with 2 n ≤ κ .We write rank A for the rank of a complex matrix A and det B for the determinant of asquare complex matrix B . Remark . Let n ∈ N and let ( s j ) nj =0 ∈ H < q, n . Then rank H n = P nk =0 rank L k and det H n = Q nk =0 det L k . Notation . Let ( s j ) κj =0 be a sequence of complex p × q matrices. Then let Θ := O p × q andΘ n := z n, n − H † n − y n, n − for all n ∈ N with 2 n − ≤ κ .7 efinition 3.18. If ( s j ) κj =0 is a sequence of complex p × q matrices, then (using Notation 3.2)the sequences ( a j ) κj =0 and ( b j ) κj =0 given by a k := αs k + Θ α,k, • and b k := βs k − Θ • ,k,β for all k ∈ N with 2 k ≤ κ and by a k +1 := Θ k +1 and b k +1 := − αβs k + ( α + β ) s k +1 − Θ α,k,β for all k ∈ N with 2 k + 1 ≤ κ are called the sequence of left matricial interval endpoints associatedwith ( s j ) κj =0 and [ α, β ] and the sequence of right matricial interval endpoints associated with ( s j ) κj =0 and [ α, β ], resp.By virtue of Notation 3.17, we have in particular a = αs , b = βs , a = s s † s , and b = − αβs + ( α + β ) s . (3.5)Using Lemma 3.11 and Remark A.14, we easily obtain: Remark . If ( s j ) κj =0 ∈ F < q,κ,α,β , then { a j , b j } ⊆ C q × q H for all j ∈ Z ,κ .Observe that for arbitrarily given Hermitian q × q matrices A and B , the (closed) matricialinterval [[ A, B ]] := { X ∈ C q × q H : A X B } (3.6)is non-empty if and only if A B . Theorem 3.20 ( [27, Thm. 11.2(a)]) . If m ∈ N and ( s j ) mj =0 ∈ F < q,m,α,β , then the matricialinterval [[ a m , b m ]] is non-empty and coincides with the set of all complex q × q matrices s m +1 for which ( s j ) m +1 j =0 belongs to F < q,m +1 ,α,β . Definition 3.21.
If ( s j ) κj =0 is a sequence of complex p × q matrices, then we call ( d j ) κj =0 givenby d j := b j − a j the sequence of [ α, β ] -interval lengths associated with ( s j ) κj =0 .By virtue of (3.5), we have in particular d = δs and d = − αβs + ( α + β ) s − s s † s . (3.7) Remark . Let ( s j ) κj =0 be a sequence of complex p × q matrices with sequence of[ α, β ]-interval lengths ( d j ) κj =0 . For each k ∈ Z ,κ , the matrix d k is built from the matrices s , s , . . . , s k . In particular, for each m ∈ Z ,κ , the sequence of [ α, β ]-interval lengths associ-ated with ( s j ) mj =0 coincides with ( d j ) mj =0 . Remark . Suppose κ ≥
1. If ( s j ) κj =0 ∈ D p × q,κ , then d = a s † b and d = b s † a . Definition 3.24.
Let ( s j ) κj =0 be a sequence of complex p × q matrices. Then the sequence( A j ) κj =0 given by A := s and by A j := s j − a j − is called the sequence of lower Schurcomplements associated with ( s j ) κj =0 and [ α, β ]. Furthermore, if κ ≥
1, then the sequence( B j ) κj =1 given by B j := b j − − s j is called the sequence of upper Schur complements associatedwith ( s j ) κj =0 and [ α, β ].Because of (3.5), we have in particular A = a , B = b , and B = c . (3.8) Remark . Let ( s j ) κj =0 be a sequence of complex p × q matrices. Then A n = L n for all n ∈ N with 2 n ≤ κ and A n +1 = L α,n, • for all n ∈ N with 2 n + 1 ≤ κ . In particular, if n ≥
1, then A n is the Schur complement of H n − in H n and A n +1 is the Schur complementof H α,n − , • in H α,n, • . Furthermore, B n +1 = L • ,n,β for all n ∈ N with 2 n + 1 ≤ κ and B n +2 = L α,n,β for all n ∈ N with 2 n + 2 ≤ κ . In particular, if n ≥
1, then B n +1 is the Schurcomplement of H • ,n − ,β in H • ,n,β and B n +2 is the Schur complement of H α,n − ,β in H α,n,β .8f A and B are two complex p × q matrices, then the matrix A ⊤−⊥ B := A ( A + B ) † B (3.9)is called the parallel sum of A and B . Proposition 3.26 ( [27, Thm. 10.14]) . If ( s j ) κj =0 ∈ F < q,κ,α,β , then d = δ A and furthermore d k = δ ( A k ⊤−⊥ B k ) and d k = δ ( B k ⊤−⊥ A k ) for all k ∈ Z ,κ . Proposition 3.27 ( [27, Prop. 10.15(a)]) . If ( s j ) κj =0 ∈ F < q,κ,α,β , then d j ∈ C q × q < for all j ∈ Z ,κ . Proposition 3.28 ( [27, Prop. 10.18]) . Let ( s j ) κj =0 ∈ F < q,κ,α,β . Then R ( d ) = R ( A ) and N ( d ) = N ( A ) . Furthermore, R ( d j ) = R ( A j ) ∩ R ( B j ) and N ( d j ) = N ( A j ) + N ( B j ) forall j ∈ Z ,κ , and R ( d j ) = R ( A j +1 ) + R ( B j +1 ) and N ( d j ) = N ( A j +1 ) ∩ N ( B j +1 ) for all j ∈ Z ,κ − . The ranks of the matrices considered in Proposition 3.28 are connected by means of thewell-known formula for the dimension of the sum of two arbitrary finite-dimensional linearsubspaces:
Remark . If U and U are finite-dimensional linear subspaces of some vector space, thendim( U + U ) = dim U + dim U − dim( U ∩ U ). Corollary 3.30.
Let ( s j ) κj =0 ∈ F < q,κ,α,β . Then rank d = rank A and rank d j − + rank d j =rank A j + rank B j for all j ∈ Z ,κ .Proof. From Proposition 3.28 we obtain rank d = rank A and, for all j ∈ Z ,κ , furthermorerank d j − = dim( R ( A j ) + R ( B j )) and rank d j = dim( R ( A j ) ∩ R ( B j )). The application ofRemark 3.29 to the linear subspaces R ( A j ) and R ( B j ) of the finite-dimensional vector space C q yields then rank d j − = rank A j + rank B j − rank d j .Using Corollary 3.30, we are able to derive certain relations between the ranks of the matrices d j and the ranks of the underlying block Hankel matrices: Lemma 3.31.
Let ( s j ) κj =0 ∈ F < q,κ,α,β . Then rank d = rank H . Furthermore, P n +1 ℓ =0 rank d ℓ =rank H α,n, • + rank H • ,n,β for all n ∈ N with n + 1 ≤ κ and P nℓ =0 rank d ℓ = rank H n +rank H α,n − ,β for all n ∈ N with n ≤ κ .Proof. Because of H = s = L = A and Corollary 3.30, we have rank d = rank H . Nowconsider an arbitrary n ∈ N with 2 n + 1 ≤ κ . From Proposition 3.8 and (3.2) we see that thesequences ( a j ) nj =0 and ( b j ) nj =0 both belong to H < q, n . Thus, we apply Remark 3.16 to obtainrank H α,n, • = P nk =0 rank L α,k, • and rank H • ,n,β = P nk =0 rank L • ,k,β . Using Corollary 3.30 andRemark 3.25, we get then n +1 X ℓ =0 rank d ℓ = n X k =0 (rank d k + rank d k +1 ) = n X k =0 (rank A k +1 + rank B k +1 )= n X k =0 rank A k +1 + n X k =0 rank B k +1 = rank H α,n, • + rank H • ,n,β . n ∈ N with 2 n ≤ κ . From Proposition 3.8 and (3.1) we infer( s j ) nj =0 ∈ H < q, n and ( c j ) n − j =0 ∈ H < q, n − . Thus, Remark 3.16 yields rank H n = P nk =0 rank L k and rank H α,n − ,β = P n − k =0 rank L α,k,β . Using Corollary 3.30 and Remark 3.25, we get then n X ℓ =0 rank d ℓ = rank d + n X m =1 (rank d m − + rank d m ) = rank A + n X m =1 (rank A m + rank B m )= n X k =0 rank A k + n − X k =0 rank B k +2 = rank H n + rank H α,n − ,β . Proposition 3.32 ( [27, Cor. 10.21]) . Let ( s j ) κj =0 ∈ F < q,κ,α,β and assume κ ≥ . For all j ∈ Z ,κ , then d j = δ A j d † j − B j and d j = δ B j d † j − A j . Corollary 3.33. If ( s j ) κj =0 ∈ F < q,κ,α,β , then det d = δ q det A and, for all j ∈ Z ,κ , further-more det d j − det d j = δ q det A j det B j . (3.10) Proof.
Because of Proposition 3.26 we have det d = δ q det A . Now assume κ ≥ j ∈ Z ,κ . First we consider the case that det d j − = 0. From Proposition 3.28 we can infer N ( d j − ) ⊆ N ( A j ). Consequently, det A j = 0 follows. Hence, (3.10) is fulfilled. Now weconsider the case det d j − = 0. In view of Remark A.13, then (3.10) is a consequence ofProposition 3.32. Lemma 3.34.
Let ( s j ) κj =0 ∈ F < q,κ,α,β . Then det d = δ q det H . Furthermore, Q n +1 ℓ =0 det d ℓ = δ ( n +1) q det( H α,n, • ) det( H • ,n,β ) for all n ∈ N with n + 1 ≤ κ and Q nℓ =0 det d ℓ = δ ( n +1) q det( H n ) det( H α,n − ,β ) for all n ∈ N with n ≤ κ .Proof. Because of H = s = L = A and Corollary 3.33 we have det d = δ q det H . Nowconsider an arbitrary n ∈ N with 2 n + 1 ≤ κ . With the same reasoning as in Lemma 3.31, wecan infer from Remark 3.16 then det H α,n, • = Q nk =0 det L α,k, • and det H • ,n,β = Q nk =0 det L • ,k,β .Using Corollary 3.33 and Remark 3.25, we get then n +1 Y ℓ =0 det d ℓ = n Y k =0 (det d k det d k +1 ) = n Y k =0 ( δ q det A k +1 det B k +1 )= δ ( n +1) q n Y k =0 det A k +1 n Y k =0 det B k +1 = δ ( n +1) q det H α,n, • det H • ,n,β . Now consider an arbitrary n ∈ N with 2 n ≤ κ . With the same reasoning as in Lemma 3.31,we can conclude from Remark 3.16 analogously det H n = Q nk =0 det L k and det H α,n − ,β = Q n − k =0 det L α,k,β . Using Corollary 3.33 and Remark 3.25, we obtain then n Y ℓ =0 det d ℓ = det d n Y m =1 (det d m − det d m ) = δ q det A n Y m =1 ( δ q det A m det B m )= δ ( n +1) q n Y k =0 det A k n − Y k =0 det B k +2 = δ ( n +1) q det H n det H α,n − ,β . Corollary 3.35 (cf. [28, Cor. 5.25]) . Let m ∈ N , let ( s j ) mj =0 ∈ F < q,m,α,β , let λ ∈ [0 , ,and let s m +1 := a m + λ d m . Then, the sequence ( s j ) m +1 j =0 belongs to F < q,m +1 ,α,β . Furthermore, A m +1 = λ d m , B m +1 = (1 − λ ) d m , and d m +1 = δλ (1 − λ ) d m . In [28, Def. 6.1], we subsumed the Schur complements mentioned in Remark 3.25 to aparameter sequence:
Definition 3.36.
Let ( s j ) κj =0 be a sequence of complex p × q matrices. Let the sequence( f j ) κj =0 be given by f := A , by f k +1 := A k +1 and f k +2 := B k +1 for all k ∈ N with2 k + 1 ≤ κ , and by f k +3 := B k +2 and f k +4 := A k +2 for all k ∈ N with 2 k + 2 ≤ κ . Thenwe call ( f j ) κj =0 the F α,β -parameter sequence of ( s j ) κj =0 .In view of (3.8) and (3.5), we have in particular f = s , f = a = s − αs , and f = b = βs − s . (3.11)The [ α, β ]-non-negative definiteness as well as rank constellations among the non-negativeHermitian block Hankel matrices H n , H α,n, • , H • ,n,β , and H α,n,β can be characterized in termsof F α,β -parameters (cf. [28, Propositions 6.13 and 6.14]). Remark . Let ( s j ) κj =0 be a sequence of complex p × q matrices with F α,β -parameter se-quence ( f j ) κj =0 . Then f = s . Furthermore, for each k ∈ Z ,κ , the matrices f k − and f k arebuilt from the matrices s , s , . . . , s k . In particular, for each m ∈ Z ,κ , the F α,β -parametersequence of ( s j ) mj =0 coincides with ( f j ) mj =0 . Proposition 3.38 ( [28, Prop. 6.14]) . Let ( s j ) κj =0 be a sequence of complex q × q matrices.Then ( s j ) κj =0 ∈ F < q,κ,α,β if and only if f j ∈ C q × q < for all j ∈ Z , κ .Remark . Let ( s j ) κj =0 be a sequence of complex p × q matrices. For all k ∈ Z ,κ , then f k − = d k − − f k .To single out all sequences ( f j ) κj =0 of complex q × q matrices which indeed occur as F α,β -parameters of sequences ( s j ) κj =0 ∈ F < q,κ,α,β , we introduced in [28, Notation 6.19] thefollowing class: Notation . For each η ∈ [0 , ∞ ), denote by C < q,κ,η the set of all sequences ( f j ) κj =0 of non-negative Hermitian q × q matrices satisfying, in the case κ ≥
1, the equations ηf = f + f and η ( f k − ⊤−⊥ f k ) = f k +1 + f k +2 for all k ∈ Z ,κ − . Theorem 3.41 (cf. [28, Thm. 6.20]) . The mapping Γ α,β : F < q,κ,α,β → C < q,κ,δ given by ( s j ) κj =0 ( f j ) κj =0 is well defined and bijective. For each matrix A ∈ C q × q < , there exists a uniquely determined matrix Q ∈ C q × q < with Q = A called the non-negative Hermitian square root Q = A / of A . To uncover relations betweenthe F α,β -parameters ( f j ) κj =0 and to obtain a parametrization of the set F < q,κ,α,β , we introducedin [28, Def. 6.21 and Notation 6.28] another parameter sequence ( e j ) κj =0 and a correspondingclass E < q,κ,δ of sequences of complex matrices. (Observe that these constructions are well defineddue to Proposition 3.27 and Remark A.25.) 11 efinition 3.42. Let ( s j ) κj =0 ∈ F < q,κ,α,β with F α,β -parameter sequence ( f j ) κj =0 and sequenceof [ α, β ]-interval lengths ( d j ) κj =0 . Then we call ( e j ) κj =0 given by e := f and by e j :=( d / j − ) † f j ( d / j − ) † for each j ∈ Z ,κ the [ α, β ] -interval parameter sequence of ( s j ) κj =0 . Lemma 3.43 (cf. [28, Prop. 6.27]) . If ( s j ) κj =0 ∈ F < q,κ,α,β , then f j = d / j − e j d / j − for all j ∈ Z ,κ . With the Euclidean scalar product h· , ·i E : C q × C q → C given by h x, y i E := y ∗ x , which is C -linear in its first argument, the C -vector space C q becomes a unitary space. Let U be anarbitrary non-empty subset of C q . The orthogonal complement U ⊥ := { v ∈ C q : h v, u i E =0 for all u ∈ U } of U is a linear subspace of the unitary space C q . If U is a linear subspaceitself, the unitary space C q is the orthogonal sum of U and U ⊥ . In this case, we write P U forthe transformation matrix corresponding to the orthogonal projection onto U with respect tothe standard basis of C q , i. e., P U is the uniquely determined matrix P ∈ C q × q satisfying thethree conditions P = P , P ∗ = P , and R ( P ) = U . Notation . For each η ∈ [0 , ∞ ), let E < q,κ,η be the set of all sequences ( e k ) κk =0 from C q × q < which fulfill the following condition: If κ ≥
1, then e k P R ( d k − ) for all k ∈ Z ,κ , where thesequence ( d k ) κk =0 is recursively given by d := ηe and d k := ηd / k − e / k ( P R ( d k − ) − e k ) e / k d / k − . Regarding Theorem 3.20, in the case q = 1 (cf. [12, Sec. 1.3]), the (classical) canonicalmoments p , p , p , . . . of a point in the moment space corresponding to a probability measure µ on [ α, β ] = [0 ,
1] are given in our notation by p k = s k − a k − b k − − a k − = A k d k − , k ∈ N , where the sequence ( s j ) ∞ j =0 of power moments s j := R [0 , x j µ (d x ) associated with µ is [0 , s = 1. Observe that the [0 , e j ) ∞ j =0 of ( s j ) ∞ j =0 areconnected to the canonical moments via p = 1 − e , p = e , p = 1 − e , p = e , p = 1 − e , . . . (3.12)The quantities q k = 1 − p k occur in the classical framework as well (see, e. g. [12, Sec. 1.3]). Inthe general case q ∈ N we have the following: Theorem 3.45 ( [28, Thm. 6.30]) . The mapping Σ α,β : F < q,κ,α,β → E < q,κ,δ given by ( s j ) κj =0 ( e j ) κj =0 is well defined and bijective. Proposition 3.46 (cf. [28, Prop. 6.32]) . Let ( s j ) κj =0 ∈ F < q,κ,α,β . Then e < O q × q and e j ∈ [[ O q × q , P R ( d j − ) ]] for all j ∈ Z ,κ . Furthermore, d = δ e and d j = δ d / j − e / j ( P R ( d j − ) − e j ) e / j d / j − for all j ∈ Z ,κ . Example 3.47 (cf. [29, Example 7.35]) . Let λ ∈ (0 , B ∈ C q × q < , and let P := P R ( B ) .Then ( e j ) ∞ j =0 given by e := B and by e j := λP for all j ∈ N is the [ α, β ]-interval parametersequence of a sequence ( s j ) ∞ j =0 ∈ F < q, ∞ ,α,β .We continue by recalling the construction of a certain transformation for sequences of matri-ces. This transformation was introduced in [29] and constitutes the elementary step of a Schurtype algorithm in the class of [ α, β ]-non-negative definite sequences, i. e., Hausdorff momentsequences: 12 efinition 3.48. Let ( s j ) κj =0 be a sequence of complex p × q matrices. Further let b − := − s and, in the case κ ≥
1, let ( b j ) κ − j =0 be given by Notation 3.2. Then we call the sequence ( b j ) κj =0 given by b j := b j − the ( −∞ , β ] -modification of ( s j ) κj =0 .In particular, if β = 0, then ( b j ) κj =0 coincides with the sequence ( − s j ) κj =0 . For an arbitrary β ∈ R , the sequence ( s j ) κj =0 is reconstructible from ( b j ) κj =0 as well.Let ( s j ) κj =0 and ( t j ) κj =0 be sequences of complex p × q and q × r matrices, resp. As usual,the Cauchy product ( x j ) κj =0 of ( s j ) κj =0 and ( t j ) κj =0 is given by x j := P jℓ =0 s ℓ t j − ℓ . Definition 3.49.
Let ( s j ) κj =0 be a sequence of complex p × q matrices. Then we call thesequence ( s ♯j ) κj =0 defined by s ♯ := s † and, for all j ∈ Z ,κ , recursively by s ♯j := − s † P j − ℓ =0 s j − ℓ s ♯ℓ the reciprocal sequence associated to ( s j ) κj =0 . Remark . Let ( s j ) κj =0 be a sequence of complex p × q matrices with reciprocal sequence( r j ) κj =0 . For each k ∈ Z ,κ , then the matrix r k is built from the matrices s , s , . . . , s k . Inparticular, for each m ∈ Z ,κ , the reciprocal sequence associated to ( s j ) mj =0 coincides with( r j ) mj =0 .Using the Cauchy product and the reciprocal sequence associated to the( −∞ , β ]-modification of ( a j ) κ − j =0 , we introduce now a transformation of sequences ofcomplex matrices: Definition 3.51.
Suppose κ ≥
1. Let ( s j ) κj =0 be a sequence of complex p × q matrices. Denoteby ( g j ) κ − j =0 the ( −∞ , β ]-modification of ( a j ) κ − j =0 and by ( x j ) κ − j =0 the Cauchy product of ( b j ) κ − j =0 and ( g ♯j ) κ − j =0 . Then we call the sequence ( t j ) κ − j =0 given by t j := − a s † x j a the F α,β -transformof ( s j ) κj =0 .Since, in the classical case that α = 0 and β = 1, the sequence ( a j ) κ − j =0 coincides with theshifted sequence ( s j +1 ) κ − j =0 , the F , -transform is given by t j = − s s † x j s with the Cauchyproduct ( x j ) κ − j =0 of ( b j ) κ − j =0 and ( g ♯j ) κ − j =0 , where the sequence ( b j ) κ − j =0 is given by b j = s j − s j +1 and the sequence ( g j ) κ − j =0 is given by g = − s and by g j = s j − s j +1 for j ∈ Z ,κ − . Remark . Assume κ ≥
1. Let ( s j ) κj =0 be a sequence of complex p × q matrices with F α,β -transform ( t j ) κ − j =0 . Then one can see from Remark 3.50 that, for each k ∈ Z ,κ − , thematrix t k is built from the matrices s , s , . . . , s k +1 . In particular, for all m ∈ Z ,κ , the F α,β -transform of ( s j ) mj =0 coincides with ( t j ) m − j =0 . Lemma 3.53 ( [29, Lem. 8.29]) . Suppose κ ≥ . Let ( s j ) κj =0 ∈ D p × q,κ . Denote by ( t j ) κ − j =0 the F α,β -transform of ( s j ) κj =0 . Then t = d , where d is given by Definition 3.21. Using the parallel sum given via (3.9), the effect caused by F α,β -transformation on the F α,β -parameters can be completely described: Corollary 3.54 ( [29, Cor. 9.10]) . Assume κ ≥ and let ( s j ) κj =0 ∈ F < q,κ,α,β with F α,β -transform ( t j ) κ − j =0 . Denote by ( g j ) κ − j =0 the F α,β -parameter sequence of ( t j ) κ − j =0 . Then g = δ ( f ⊤−⊥ f ) and g j = δ f j +2 for all j ∈ Z , κ − . We are now going to iterate the F α,β -transformation:13 efinition 3.55. Let ( s j ) κj =0 be a sequence of complex p × q matrices. Let the sequence( s { } j ) κj =0 be given by s { } j := s j . If κ ≥
1, then, for all k ∈ Z ,κ , let the sequence ( s { k } j ) κ − kj =0 berecursively defined to be the F α,β -transform of the sequence ( s { k − } j ) κ − ( k − j =0 . For all k ∈ Z ,κ ,then we call the sequence ( s { k } j ) κ − kj =0 the k -th F α,β -transform of ( s j ) κj =0 . Remark . Suppose κ ≥
1. Let ( s j ) κj =0 be a sequence of complex p × q matrices. Then( s { } j ) κ − j =0 is exactly the F α,β -transform of ( s j ) κj =0 from Definition 3.55. Remark . Let k ∈ Z ,κ and let ( s j ) κj =0 be a sequence of complex p × q matrices with k -th F α,β -transform ( u j ) κ − kj =0 . In view of Remark 3.52, we see that, for each ℓ ∈ Z ,κ − k , the matrix u ℓ is built only from the matrices s , s , . . . , s ℓ + k . In particular, for each m ∈ Z k,κ , the k -th F α,β -transform of ( s j ) mj =0 coincides with ( u j ) m − kj =0 . Proposition 3.58 ( [29, Thm. 9.4]) . If ( s j ) κj =0 ∈ F < q,κ,α,β , then ( s { k } j ) κ − kj =0 ∈ F < q,κ − k,α,β forall k ∈ Z ,κ . Proposition 3.59 ( [29, Prop. 9.8]) . Let ( s j ) κj =0 ∈ F < q,κ,α,β with F α,β -parameter sequence ( f j ) κj =0 . Then f = s and furthermore f k +1 = δ − k a { k } and f k +2 = δ − k b { k } for all k ∈ N with k + 1 ≤ κ and f k +3 = δ − (2 k +1) a { k +1 } and f k +4 = δ − (2 k +1) b { k +1 } for all k ∈ N with k + 2 ≤ κ . In the following result, we express the sequence of [ α, β ]-interval lengths introduced in Defi-nition 3.21 by the F α,β -transforms of an [ α, β ]-non-negative definite sequence: Corollary 3.60 ( [29, Cor. 9.9]) . If ( s j ) κj =0 ∈ F < q,κ,α,β , then d j = δ − ( j − s { j } for all j ∈ Z ,κ . Proposition 3.61 ( [29, Prop. 9.11]) . Let k ∈ Z ,κ and let ( s j ) κj =0 ∈ F < q,κ,α,β with sequence of [ α, β ] -interval lengths ( d j ) κj =0 and k -th F α,β -transform ( s { k } j ) κ − kj =0 . Then ( δ k d k + j ) κ − kj =0 coincideswith the sequence of [ α, β ] -interval lengths associated with ( s { k } j ) κ − kj =0 . Theorem 3.62 ( [29, Thm. 9.14]) . Let ( s j ) κj =0 ∈ F < q,κ,α,β with sequence of [ α, β ] -intervallengths ( d j ) κj =0 and [ α, β ] -interval parameter sequence ( e j ) κj =0 . Let k ∈ Z ,κ . Then the k -th F α,β -transform ( s { k } j ) κ − kj =0 of ( s j ) κj =0 belongs to F < q,κ − k,α,β and the [ α, β ] -interval parametersequence ( p j ) κ − kj =0 of ( s { k } j ) κ − kj =0 fulfills p = δ k − d k and p j = e k + j for all j ∈ Z ,κ − k . Now we look for a characterization of the fixed points of the F α,β -transformation. Corollary 3.63.
Suppose κ ≥ . Let ( s j ) κj =0 ∈ F < q,κ,α,β with [ α, β ] -interval parameter sequence ( e j ) κj =0 and F α,β -transform ( t j ) κ − j =0 . Then t = s if and only if e − e = δ − P R ( s ) .Proof. Proposition 3.46 yields d = δ e and d = δ d / e / ( P R ( d ) − e ) e / d / . Because ofLemma 3.11, the matrix s is non-negative Hermitian. According to (3.7), we have d = δs .In view of δ >
0, then d / = δ / s / as well as R ( d ) = R ( s ) and N ( d ) = N ( s ) follow.Consequently, we can conclude d = δ s / e / ( P R ( s ) − e ) e / s / . Using Remark A.14, wecan infer from Definition 3.42 furthermore R ( e ) ⊆ R (( d / ) † ) = R (( d / ) ∗ ) = R ( d / ) = R ( d ) = R ( s ) and, similarly, N ( s ) ⊆ N ( e ). In particular, R ( e / ) = R ( e ) ⊆ R ( s ) follows,14mplying e / P R ( s ) = e / . Hence, d = δ s / ( e − e ) s / . Therefore, R ( d ) ⊆ R ( s / ) and N ( s / ) ⊆ N ( d ). In view of Remark 3.10, the application of Lemma 3.53 yields t = d .Hence, it remains to show that δ s / ( e − e ) s / = s is equivalent to e − e = δ − P R ( s ) .First assume that δ s / ( e − e ) s / = s is fulfilled. Taking into account Remark A.14,we see R ( e ) ⊆ R ( s ) = R ( s / ) = R (( s / ) ∗ ) = R (( s / ) † ) and similarly N (( s / ) † ) ⊆N ( e ). Thus, we can infer from Remarks A.20, A.21, and A.14 that δ ( e − e ) =( s / ) † s ( s / ) † . Because of R ( s ) = R ( s / ) = R (( s / ) ∗ ), the application of Remark A.18yields ( s / ) † s ( s / ) † = P R (( s / ) ∗ ) P R ( s / ) = P R ( s ) = P R ( s ) . Consequently, e − e = δ − P R ( s ) .Conversely, assume that e − e = δ − P R ( s ) holds true. In view of R ( s / ) = R ( s ), we have s / P R ( s ) = s / . Thus, we obtain δ s / ( e − e ) s / = s / P R ( s ) s / = s . Corollary 3.64.
Let ( s j ) ∞ j =0 ∈ F < q, ∞ ,α,β with [ α, β ] -interval parameter sequence ( e j ) ∞ j =0 and F α,β -transform ( t j ) ∞ j =0 . Then the following statements are equivalent:(i) ( t j ) ∞ j =0 coincides with ( s j ) ∞ j =0 , i. e. the sequence ( s j ) ∞ j =0 is a fixed point of the F α,β -transformation.(ii) e − e = δ − P R ( s ) and e j = e for all j ∈ N .Proof. In view of Remark 3.56, we see from Theorem 3.62 that ( t j ) ∞ j =0 belongs to F < q, ∞ ,α,β andthat the [ α, β ]-interval parameter sequence ( p j ) ∞ j =0 of ( t j ) ∞ j =0 fulfills p = d and p j = e j +1 forall j ∈ N . Because of Theorem 3.45, statement (i) holds if and only if p j = e j for all j ∈ N .Consequently, (i) is fulfilled if and only if d = e and e j +1 = e j for all j ∈ N . Hence, itremains to show that d = e is equivalent to e − e = δ − P R ( s ) . In view of Remark 3.10,the application of Lemma 3.53 yields t = d . By virtue of Definition 3.42 and (3.11), we have e = f = s . The application of Corollary 3.63 completes the proof.Now we draw special attention to the scalar case q = 1. Example 3.65.
Let Φ : F < , ∞ ,α,β → F < , ∞ ,α,β be defined by ( s j ) ∞ j =0 ( t j ) ∞ j =0 , where ( t j ) ∞ j =0 isthe F α,β -transform of ( s j ) ∞ j =0 . Then:(a) If δ <
2, then Φ has one single fixed point ( s j ) ∞ j =0 given by s j = 0.(b) Suppose δ = 2. For each M ∈ [0 , ∞ ) there exists a unique fixed point ( s M ; j ) ∞ j =0 of Φwith s M ;0 = M . If M = 0, then ( s M ; j ) ∞ j =0 is given by s M ; j = 0. If M >
0, then ( s M ; j ) ∞ j =0 corresponds to the [ α, β ]-interval parameter sequence ( e M ; j ) ∞ j =0 given by e M ;0 = M andby e M ; j = for j ∈ N .(c) Suppose δ >
2. Then Φ has exactly one fixed point ( s j ) ∞ j =0 with s = 0, namely( s j ) ∞ j =0 given by s j = 0. For each M ∈ (0 , ∞ ), furthermore Φ has exactly two fixedpoints ( s ± M ; j ) ∞ j =0 with s ± M ;0 = M , namely ( s ± M ; j ) ∞ j =0 corresponding to the [ α, β ]-intervalparameter sequences ( e ± M ; j ) ∞ j =0 given by e ± M ;0 = M and by e ± M ; j = ± δ √ δ − j ∈ N . 15 roof. (a) Assume δ <
2. It is readily checked that the sequence ( s j ) ∞ j =0 defined by s j := 0belongs to F < , ∞ ,α,β and that the sequence ( e j ) ∞ j =0 defined by e j := 0 is the [ α, β ]-intervalparameter sequence of ( s j ) ∞ j =0 . In view of P R ( s ) = 0, we can thus infer from Corollary 3.64that ( s j ) ∞ j =0 is a fixed point of Φ.Now consider an arbitrary fixed point ( s j ) ∞ j =0 ∈ F < , ∞ ,α,β of Φ. If s = 0, then from Re-mark 3.10 we can conclude that s j = 0 for all j ∈ N , i. e., ( s j ) ∞ j =0 coincides with ( s j ) ∞ j =0 .Consider now the case s = 0. Then P R ( s ) = 1. Denote by ( e j ) ∞ j =0 the [ α, β ]-interval parame-ter sequence of ( s j ) ∞ j =0 . Because of Corollary 3.64, we have e − e = δ − P R ( s ) . Consequently,0 < δ − = δ − P R ( s ) = e − e = − ( e − ) ≤ , contradicting δ <
2. Thus, ( s j ) ∞ j =0 is theonly fixed point of Φ.(b) As above, we see that ( s j ) ∞ j =0 ∈ F < , ∞ ,α,β is a fixed point of Φ with [ α, β ]-intervalparameter sequence ( e j ) ∞ j =0 given by e j = 0. Consider an arbitrary M ∈ (0 , ∞ ) and let( e M ; j ) ∞ j =0 be defined by e M ;0 := M and by e M ; j := for j ∈ N . In view of P R ( M ) = 1, dueto Example 3.47 there exists a sequence ( s M ; j ) ∞ j =0 ∈ F < , ∞ ,α,β with [ α, β ]-interval parametersequence ( e M ; j ) ∞ j =0 . According to Theorem 3.45, the sequences ( s j ) ∞ j =0 and ( s M ; j ) ∞ j =0 aredifferent. By virtue of Definition 3.42 and (3.11), we have M = e M ;0 = s M ;0 . In particular P R ( s M ;0 ) = 1. Taking additionally into account e M ;1 = and δ = 2, hence e M ;1 − e M ;1 = = δ − P R ( s M ;0 ) follows. According of Corollary 3.64, then ( s M ; j ) ∞ j =0 is a fixed point of Φ.Now consider an arbitrary fixed point ( s j ) ∞ j =0 ∈ F < , ∞ ,α,β of Φ. Because of Lemma 3.11, wehave s ∈ [0 , ∞ ). If s = 0, then, as above, the sequence ( s j ) ∞ j =0 coincides with ( s j ) ∞ j =0 . Nowassume s >
0. Then P R ( s ) = 1. Denote by ( e j ) ∞ j =0 the [ α, β ]-interval parameter sequence of( s j ) ∞ j =0 . Because of Corollary 3.64, we have e − e = δ − P R ( s ) and e = e = e = · · · Takingadditionally into account δ = 2, then ( e − ) = e − e + = − δ − P R ( s ) + = − − + = 0follows, i. e. e = . By virtue of Definition 3.42 and (3.11), we have e = f = s . Setting M := s , then M ∈ (0 , ∞ ) and ( e j ) ∞ j =0 coincides with ( e M ; j ) ∞ j =0 defined as above, implyingthat ( s j ) ∞ j =0 coincides with ( s M ; j ) ∞ j =0 , according to Theorem 3.45.(c) As above, we see that ( s j ) ∞ j =0 ∈ F < , ∞ ,α,β is a fixed point of Φ with [ α, β ]-intervalparameter sequence ( e j ) ∞ j =0 given by e j = 0. Consider an arbitrary M ∈ (0 , ∞ ) and let( e ± M ; j ) ∞ j =0 be defined by e ± M ;0 := M and by e ± M ; j := ± δ √ δ − j ∈ N . In view of δ > δ − > < e − M ; j < < e + M ; j < j ∈ N . Regarding P R ( M ) =1, hence, due to Example 3.47, there exist sequences ( s − M ; j ) ∞ j =0 and ( s + M ; j ) ∞ j =0 belonging to F < , ∞ ,α,β with [ α, β ]-interval parameter sequence ( e − M ; j ) ∞ j =0 and ( e + M ; j ) ∞ j =0 , resp. According toTheorem 3.45, the sequences ( s j ) ∞ j =0 , ( s − M ; j ) ∞ j =0 , and ( s + M ; j ) ∞ j =0 are pairwise different. Byvirtue of Definition 3.42 and (3.11), we have M = e ± M ;0 = s ± M ;0 . In particular P R ( s ± M ;0 ) = 1.Taking additionally into account ( e ± M ;1 − ) = δ − δ = − δ − , hence e ± M ;1 − ( e ± M ;1 ) = − ( e ± M ;1 − ) = δ − P R ( s ± M ;0 ) follows. According of Corollary 3.64, then the sequences ( s − M ; j ) ∞ j =0 and ( s + M ; j ) ∞ j =0 are both fixed points of Φ.Now consider an arbitrary fixed point ( s j ) ∞ j =0 ∈ F < , ∞ ,α,β of Φ. Because of Lemma 3.11,we have s ∈ [0 , ∞ ). If s = 0, then, as above, the sequence ( s j ) ∞ j =0 coincides with ( s j ) ∞ j =0 .Assume s >
0. Then P R ( s ) = 1. Denote by ( e j ) ∞ j =0 the [ α, β ]-interval parameter sequence of( s j ) ∞ j =0 . Because of Corollary 3.64, we get e − e = δ − P R ( s ) and e = e = e = · · · Then( e − ) = e − e + = − δ − P R ( s ) + = − δ + = δ − δ follows, i. e. e = − δ √ δ − e = + δ √ δ −
4. By virtue of Definition 3.42 and (3.11), we have e = f = s .16etting M := s , then M ∈ (0 , ∞ ) and ( e j ) ∞ j =0 coincides with ( e − M ; j ) ∞ j =0 or with ( e + M ; j ) ∞ j =0 defined above, implying that ( s j ) ∞ j =0 coincides with ( s − M ; j ) ∞ j =0 or with ( s + M ; j ) ∞ j =0 , according toTheorem 3.45. Remark . Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Then the following statements areequivalent:(i) ( s j ) mj =0 ∈ F ≻ q,m,α,β .(ii) d j ∈ C q × q ≻ for all j ∈ Z ,m .(iii) det d j = 0 for all j ∈ Z ,m .(iv) d m ∈ C q × q ≻ .(v) det d m = 0.Indeed, in view of Proposition 3.27 we see that (ii) and (iii) resp. (iv) and (v) are equivalent.Furthermore, for each j ∈ Z ,m , from [27, Prop. 10.23] we obtain d j < (4 /δ ) m − j d m . Thus, (iv)implies (ii). The equivalence of (iii) and (i) is a consequence of Lemma 3.34. Remark . Let ( s j ) ∞ j =0 ∈ F < q, ∞ ,α,β . In view of Remark 3.66, then the following statementsare equivalent:(i) ( s j ) ∞ j =0 ∈ F ≻ q, ∞ ,α,β .(ii) d j ∈ C q × q ≻ for all j ∈ N .(iii) det d j = 0 for all j ∈ N .
4. The class R q ( C \ [ α, β ]) In this section, we introduce several classes of matrix-valued functions, holomorphic in the senseexplained in Appendix F. We consider the following open half-planes in the complex plane: H − ( α ) := { z ∈ C : Re z < α } , H + ( β ) := { z ∈ C : Re z > β } , Π − := { z ∈ C : Im z < } , andΠ + := { z ∈ C : Im z > } . Furthermore, we write Re A := ( A + A ∗ ) and Im A := ( A − A ∗ )for the real part and the imaginary part of a complex square matrix A , resp. Notation . Denote by R q (Π + ) the set of all matrix-valued functions F : Π + → C q × q , whichare holomorphic and satisfy Im F ( z ) ∈ C q × q < for all z ∈ Π + .By means of G ( z ) := ( F ( z ) , if Im z > F ( z )] ∗ , if Im z < , the matrix-valued functions F of the class R q (Π + ) can be extended to holomorphic matrix-valued functions G : C \ R → C q × q , which satisfy Im G ( z ) / Im z ∈ C q × q < for all z ∈ C \ R . Inthe scalar case q = 1, such a function is called R -function in [39, p. 1]. The matrix-valuedfunctions of the class R q (Π + ) are also called Herglotz functions , Nevanlinna functions or Pickfunctions . They admit a well-known integral representation, the scalar version of which canbe found, e. g., in [39, Eq. (S1.1.1)].Using the Euclidean norm k x k E := √ x ∗ x on C q corresponding to the Euclidean scalarproduct, we define the operator norm k A k S := max {k Au k E : u ∈ C q with k u k E = 1 } on C p × q induced by the Euclidean norms on C q and C p , which is also called spectral norm on C p × q .17 otation . Denote by R ,q (Π + ) the set of all F ∈ R q (Π + ) satisfying the growth conditionsup y ∈ [1 , ∞ ) y k F (i y ) k S < ∞ . Theorem 4.3 (cf. [10, Thm. 8.7]) . (a) If F ∈ R ,q (Π + ) , then there exists a unique σ ∈M < q ( R ) such that F ( z ) = Z R t − z σ (d t ) (4.1) holds true for all z ∈ Π + .(b) If σ ∈ M < q ( R ) , then F : Π + → C q × q defined via (4.1) belongs to R ,q (Π + ) . Definition 4.4.
Let F ∈ R ,q (Π + ). Then the unique measure σ ∈ M < q ( R ) such that (4.1)holds true for all z ∈ Π + is called the (matricial) spectral measure of F and is denoted by σ F .In certain situations, an upper bound for y k F (i y ) k S can be obtained, using Lemma A.26: Lemma 4.5 ( [10, Lem. 8.9]) . Let M ∈ C q × q and let F : Π + → C q × q be a holomorphicmatrix-valued function such that, for all z ∈ Π + , the matrix h M F ( z )[ F ( z )] ∗ z Im F ( z ) i is non-negativeHermitian. Then F ∈ R ,q (Π + ) with sup y ∈ (0 , ∞ ) y k F (i y ) k S ≤ k M k S and σ F ( R ) M . Now we introduce that class of holomorphic matrix-valued functions, which is relevant forthe moment problem MP [[ α, β ]; ( s j ) κj =0 , =]. Notation . Denote by R q ( C \ [ α, β ]) the set of all matrix-valued functions F : C \ [ α, β ] → C q × q which are holomorphic and satisfy the following conditions:(I) Im F ( z ) ∈ C q × q < for all z ∈ Π + .(II) F ( x ) ∈ C q × q < for all x ∈ ( −∞ , α ) and − F ( x ) ∈ C q × q < for all x ∈ ( β, ∞ ).Since such functions are holomorphic in C \ [ α, β ] with non-negative Hermitian imaginarypart in Π + , we can think of R q ( C \ [ α, β ]) as a subclass of R q (Π + ), by virtue of the identitytheorem for holomorphic functions: Remark . By means of restricting matrix-valued functions of R q ( C \ [ α, β ]) to Π + , an injec-tive mapping from R q ( C \ [ α, β ]) into R q (Π + ) is given.Lemma 4.12 will show that the above mentioned restrictions even belong to R ,q (Π + ). Sinceeach function from R q ( C \ [ α, β ]) is holomorphic in C \ [ α, β ] with Hermitian values in R \ [ α, β ],we can use the Schwarz reflection principle to obtain the following relation connecting its valueson the open upper and lower half-plane: Remark . If F ∈ R q ( C \ [ α, β ]), then [ F ( z )] ∗ = F ( z ) for all z ∈ C \ [ α, β ].The latter result can also be seen from the following integral representation: Theorem 4.9 (cf. [10, Thm. 1.1]) . (a) If F ∈ R q ( C \ [ α, β ]) , then there exists a unique ¨ σ ∈ M < q ([ α, β ]) such that F ( z ) = Z [ α,β ] t − z ¨ σ (d t ) (4.2) holds true for all z ∈ C \ [ α, β ] .(b) If ¨ σ ∈ M < q ([ α, β ]) , then F : C \ [ α, β ] → C q × q defined via (4.2) belongs to R q ( C \ [ α, β ]) . efinition 4.10. Let F ∈ R q ( C \ [ α, β ]). In view of Theorem 4.9, let ¨ σ be the uniquelydetermined measure from M < q ([ α, β ]) such that (4.2) holds true for all z ∈ C \ [ α, β ]. Then ¨ σ is called the R [ α, β ] -measure of F and is denoted by ¨ σ F .As already mentioned in Section 3, the power moments of a measure belonging to M < q ([ α, β ])exist for each non-negative integer order: Remark . If F ∈ R q ( C \ [ α, β ]), then the R [ α, β ]-measure ¨ σ F of F belongs to M < q, ∞ ([ α, β ]).In view of Remark 4.7, we infer the following relation to the class R ,q (Π + ): Lemma 4.12.
Let F ∈ R q ( C \ [ α, β ]) with R [ α, β ] -measure ¨ σ F and denote by f the restrictionof F onto Π + . Then f ∈ R ,q (Π + ) and the spectral measure σ f of f fulfills σ f ( R \ [ α, β ]) = O q × q and σ f ( B ) = ¨ σ F ( B ) for all B ∈ B [ α,β ] .Proof. Use Theorems 4.9 and 4.3.By virtue of Theorem 4.9, the following integral representations are readily checked:
Remark . Let F ∈ R q ( C \ [ α, β ]). For all z ∈ C \ [ α, β ], thenRe F ( z ) = Z [ α,β ] t − Re z | t − z | ¨ σ F (d t ) and Im F ( z ) = Z [ α,β ] Im z | t − z | ¨ σ F (d t ) . Now we state a useful characterization of the functions belonging to R q ( C \ [ α, β ]). Proposition 4.14.
Let F : C \ [ α, β ] → C q × q be holomorphic. Then F ∈ R q ( C \ [ α, β ]) if andonly if the following two conditions are fulfilled:(I) z Im F ( z ) ∈ C q × q < for all z ∈ C \ R .(II) Re F ( w ) ∈ C q × q < for all w ∈ H − ( α ) and − Re F ( w ) ∈ C q × q < for all w ∈ H + ( β ) .Proof. If F ∈ R q ( C \ [ α, β ]), then (I) and (II) are readily seen from Remark 4.13.Conversely, suppose that (I) and (II) are fulfilled. Due to (I), we have Im F ( z ) ∈ C q × q < for all z ∈ Π + . As in the proof of Lemma C.3, we can conclude from (I) that, for all x ∈ R \ [ α, β ], theequation Im F ( x ) = O q × q holds true, implying F ( x ) = Re F ( x ). Taking into account (II), wethus have F ( x ) ∈ C q × q < for all x ∈ ( −∞ , α ) and − F ( x ) ∈ C q × q < for all x ∈ ( β, ∞ ). RegardingNotation 4.6, hence F ∈ R q ( C \ [ α, β ]).As an immediate consequence of the following result, we see that the column space R ( F ( z ))and the null space N ( F ( z )) of a matrix-valued function F ∈ R q ( C \ [ α, β ]) are both indepen-dent of the argument z ∈ C \ [ α, β ]. Proposition 4.15.
Let F ∈ R q ( C \ [ α, β ]) . Then:(a) R ( F ( z )) = R (¨ σ F ([ α, β ])) and N ( F ( z )) = N (¨ σ F ([ α, β ])) for all z ∈ C \ [ α, β ] .(b) R (Im F ( z )) = R (¨ σ F ([ α, β ])) and N (Im F ( z )) = N (¨ σ F ([ α, β ])) for all z ∈ C \ R .(c) R (Re F ( w )) = R (¨ σ F ([ α, β ])) and N (Re F ( w )) = N (¨ σ F ([ α, β ])) for all w ∈ H − ( α ) ∪ H + ( β ) .Proof. In view of Theorem 4.9, this follows from Lemma C.5 applied with Ω = [ α, β ].19e recall the definitions of two well-studied classes of matrices.
Definition 4.16.
Let A be a complex q × q matrix. Then A is called EP matrix if R ( A ∗ ) = R ( A ). Furthermore, the matrix A is said to be almost definite if each x ∈ C q with x ∗ Ax = 0necessarily fulfills Ax = O q × . Denote by C q × q EP and C q × q AD the set of EP matrices and the set ofalmost definite matrices from C q × q , resp. Proposition 4.17. If F ∈ R q ( C \ [ α, β ]) , then F ( z ) ∈ C q × q AD for all z ∈ C \ [ α, β ] .Proof. In view of Theorem 4.9, this follows from Lemma C.6 applied with Ω = [ α, β ].According to Remark A.31, we have C q × q AD ⊆ C q × q EP . Hence, Proposition 4.17 implies that thevalues of a function F ∈ R q ( C \ [ α, β ]) fulfill R ([ F ( z )] ∗ ) = R ( F ( z )) for all z ∈ C \ [ α, β ], a factthat also can be seen from Proposition 4.15 in combination with Remark 4.8.By means of Theorem 4.9, a characterization of R q ( C \ [ α, β ]) in terms of the class R q (Π + )can be obtained: Proposition 4.18 (cf. [10, Lem. 3.6]) . Let F : C \ [ α, β ] → C q × q be holomorphic and let thematrix-valued functions g, h : Π + → C q × q be defined by g ( z ) := ( z − α ) F ( z ) and h ( z ) :=( β − z ) F ( z ) , resp. Then F ∈ R q ( C \ [ α, β ]) if and only if g and h both belong to R q (Π + ) . In view of Remark 4.11, we can associate to a given function from the class R q ( C \ [ α, β ])three auxiliary functions, which are intimately connected to the three sequences of complexmatrices introduced in Notation 3.2 (cf. Remark 5.10): Notation . Let F ∈ R q ( C \ [ α, β ]) with R [ α, β ]-measure ¨ σ F . Then let the functions F a , F b , F c : C \ [ α, β ] → C q × q be defined by F a ( z ) := ( z − α ) F ( z ) + ¨ σ F ([ α, β ]) , F b ( z ) := ( β − z ) F ( z ) − ¨ σ F ([ α, β ]) , and F c ( z ) := ( β − z )( z − α ) F ( z ) + ( α + β − z )¨ σ F ([ α, β ]) − Z [ α,β ] t ¨ σ F (d t ) . Proposition 4.20.
Let F ∈ R q ( C \ [ α, β ]) with R [ α, β ] -measure ¨ σ F . Then F a , F b , and F c belong to R q ( C \ [ α, β ]) and their R [ α, β ] -measures ¨ σ a , ¨ σ b , and ¨ σ c fulfill ¨ σ a ( B ) = Z B ( t − α )¨ σ F (d t ) , ¨ σ b ( B ) = Z B ( β − t )¨ σ F (d t ) , ¨ σ c ( B ) = Z B ( β − t )( t − α )¨ σ F (d t ) for all B ∈ B [ α,β ] .Proof. Because of Remark 4.11, the integrals R [ α,β ] t ¨ σ F (d t ) and R [ α,β ] t ¨ σ F (d t ) exist. Since t − α > β − t > t ∈ [ α, β ], we can thus conclude that R B ( t − α )¨ σ F (d t ), R B ( β − t )¨ σ F (d t ), and R B ( β − t )( t − α )¨ σ F (d t ) are non-negative Hermitian matrices forall B ∈ B [ α,β ] . Consequently, ¨ σ a , ¨ σ b , and ¨ σ c belong to M < q ([ α, β ]). Consider now an arbitrary z ∈ C \ [ α, β ]. In view of Theorem 4.9, we have( z − α ) F ( z ) = Z [ α,β ] z − αt − z ¨ σ F (d t ) = Z [ α,β ] (cid:18) t − αt − z − (cid:19) ¨ σ F (d t ) = Z [ α,β ] t − αt − z ¨ σ F (d t ) − ¨ σ F ([ α, β ])20nd similarly ( β − z ) F ( z ) = R [ α,β ] ( t − z ) − ( β − t )¨ σ F (d t ) + ¨ σ F ([ α, β ]). Due to Theorem 4.9,then F a and F b belong to R q ( C \ [ α, β ]) having the asserted R [ α, β ]-measures. Using therepresentation above, we obtain furthermore( β − z )( z − α ) F ( z ) = Z [ α,β ] ( β − z )( t − α ) t − z ¨ σ F (d t ) − ( β − z )¨ σ F ([ α, β ])= Z [ α,β ] (cid:18) β − tt − z + 1 (cid:19) ( t − α )¨ σ F (d t ) − ( β − z )¨ σ F ([ α, β ])= Z [ α,β ] ( β − t )( t − α ) t − z ¨ σ F (d t ) + Z [ α,β ] t ¨ σ F (d t ) − ( α + β − z )¨ σ F ([ α, β ]) . Hence, F c ( z ) = R [ α,β ] 1 t − z ¨ σ c (d t ). By virtue of Theorem 4.9, thus F c belongs to R q ( C \ [ α, β ])having the asserted R [ α, β ]-measure.The combination of Proposition 4.20 with Remarks 4.13 and A.2 yields: Remark . Let F ∈ R q ( C \ [ α, β ]). For all z ∈ C \ [ α, β ], thenIm[( z − α ) F ( z )] = Im F a ( z ) = Im( z ) Z [ α,β ] t − α | t − z | ¨ σ F (d t ) , Im[( β − z ) F ( z )] = Im F b ( z ) = Im( z ) Z [ α,β ] β − t | t − z | ¨ σ F (d t ) , and Im[( β − z )( z − α ) F ( z )] = Im F c ( z ) + Im( z )¨ σ F ([ α, β ])= Im( z ) " ¨ σ F ([ α, β ]) + Z [ α,β ] ( β − t )( t − α ) | t − z | ¨ σ F (d t ) . Using Remark 4.21, the following result is readily checked:
Remark . If F ∈ R q ( C \ [ α, β ]), then z Im[( β − z )( z − α ) F ( z )] < ¨ σ F ([ α, β ]) < O q × q forall z ∈ C \ R .
5. An equivalent problem in the class R q ( C \ [ α, β ]) To describe the solution set of moment problems on the real axis, the transition to holomorphicfunctions by means of the Stieltjes transformation considered in detail in Appendix C hasturned out to be very helpful. For the sake of a simpler description of the relation to R -functionstreated in the previous section, as usual, we choose in the case Ω = R for the Stieltjes transform S the domain Π + instead of C \ R : Definition 5.1.
Let σ ∈ M < q ( R ). Then we call the matrix-valued function S σ : Π + → C q × q defined by S σ ( z ) := Z R t − z σ (d t )the R -Stieltjes transform of σ .From Theorem 4.3 and Definitions 4.4 and 5.1 we immediately see the well-known connectionof the R -Stieltjes transformation to the class R ,q (Π + ):21 roposition 5.2. (a) If F ∈ R ,q (Π + ) , then there exists a unique σ ∈ M < q ( R ) fulfilling F = S σ , namely σ = σ F .(b) If σ ∈ M < q ( R ) , then the R -Stieltjes transform S σ of σ belongs to R ,q (Π + ) . According to our interest in the matricial Hausdorff moment problem, we consider the in-tegral transformation (C.1) for the particular case of non-negative Hermitian measures σ be-longing to M < q ([ α, β ]): Definition 5.3.
Let σ ∈ M < q ([ α, β ]). Then we call the matrix-valued function ¨ S σ : C \ [ α, β ] → C q × q defined by ¨ S σ ( z ) := Z [ α,β ] t − z σ (d t ) (5.1)the [ α, β ] -Stieltjes transform of σ .The [ α, β ]-Stieltjes transform of a non-negative Hermitian measure from M < q ([ α, β ]) admitsa power series representation at z = ∞ involving the corresponding moments: Proposition 5.4 ( [5, Satz 1.2.16, p. 34]) . Let σ ∈ M < q ([ α, β ]) . Then the moments s ( σ ) j := R [ α,β ] x j σ (d x ) exist for all j ∈ N . For each z ∈ C with | z | > max {| α | , | β |} , furthermore z ∈ C \ [ α, β ] and ¨ S σ ( z ) = − ∞ X j =0 z − ( j +1) s ( σ ) j . The following reformulation of Theorem 4.9 describes the relation between [ α, β ]-Stieltjestransform ¨ S σ and R [ α, β ]-measure ¨ σ F : Proposition 5.5.
The mapping Λ [ α,β ] : M < q ([ α, β ]) → R q ( C \ [ α, β ]) given by σ ¨ S σ ,where ¨ S σ is given by (5.1) , is well defined and bijective. Its inverse Λ − α,β ] : R q ( C \ [ α, β ]) →M < q ([ α, β ]) is given by F ¨ σ F , where ¨ σ F denotes the R [ α, β ] -measure of F. By virtue of Proposition 5.5, the moment problem MP [[ α, β ]; ( s j ) κj =0 , =] admits a reformula-tion as an equivalent problem for functions belonging to the class R q ( C \ [ α, β ]): Problem FP [[ α, β ]; ( s j ) κj =0 ] : Given a sequence ( s j ) κj =0 of complex q × q matrices, parametrizethe set R q [[ α, β ]; ( s j ) κj =0 ] of all F ∈ R q ( C \ [ α, β ]) with R [ α, β ]-measure ¨ σ F belonging to M < q,κ [[ α, β ]; ( s j ) κj =0 , =].In particular, Problem FP [[ α, β ]; ( s j ) κj =0 ] has a solution if and only if the moment prob-lem MP [[ α, β ]; ( s j ) κj =0 , =] has a solution. From Theorem 3.5 we can therefore conclude: Proposition 5.6.
Let ( s j ) κj =0 be a sequence of complex q × q matrices. Then the set R q [[ α, β ]; ( s j ) κj =0 ] is non-empty if and only if the sequence ( s j ) κj =0 belongs to F < q,κ,α,β . In view of Proposition 5.5, the solution set R q [[ α, β ]; ( s j ) κj =0 ] of Problem FP [[ α, β ]; ( s j ) κj =0 ]can also be described in the following way: Remark . R q [[ α, β ]; ( s j ) κj =0 ] = { ¨ S σ : σ ∈ M < q,κ [[ α, β ]; ( s j ) κj =0 , =] } .In combination with Propositions 3.6 and 5.4, we can infer from Remark 5.7 in particular:22 roposition 5.8. Let ( s j ) ∞ j =0 ∈ F < q, ∞ ,α,β . Then the set R q [[ α, β ]; ( s j ) ∞ j =0 ] consists of exactlyone element S . For all z ∈ C with | z | > max {| α | , | β |} , furthermore z ∈ C \ [ α, β ] and S ( z ) = − ∞ X j =0 z − ( j +1) s j . (5.2)In addition, we have the following result: Proposition 5.9.
Let ( s j ) ∞ j =0 ∈ F < q, ∞ ,α,β , let F : C \ [ α, β ] → C q × q be holomorphic, and let ρ ∈ R with ρ ≥ max {| α | , | β |} . Suppose that F ( z ) = − P ∞ j =0 z − ( j +1) s j holds true for all z ∈ C with | z | > ρ . Then F ∈ R q [[ α, β ]; ( s j ) ∞ j =0 ] .Proof. Due to Proposition 5.8, the set R q [[ α, β ]; ( s j ) ∞ j =0 ] consists of exactly one element S and(5.2) holds true for all z ∈ C with | z | > max {| α | , | β |} . In particular, F ( z ) = S ( z ) for all z ∈ C with | z | > ρ follows. Consequently, the application of the identity theorem for holomorphicfunctions yields F = S . Therefore, F belongs to R q [[ α, β ]; ( s j ) ∞ j =0 ].In view of Proposition 4.20, the sequences ( a j ) κ − j =0 , ( b j ) κ − j =0 , and ( c j ) κ − j =0 introduced in No-tation 3.2 consist of the first power moments of the R [ α, β ]-measures ¨ σ a , ¨ σ b , and ¨ σ c of thematrix-valued functions F a , F b , and F c built, according to Notation 4.19, from a given function F ∈ R q [[ α, β ]; ( s j ) κj =0 ]: Remark . Let ( s j ) κj =0 ∈ F < q,κ,α,β and let F ∈ R q [[ α, β ]; ( s j ) κj =0 ]. If κ ≥
1, then F a ∈R q [[ α, β ]; ( a j ) κ − j =0 ] and F b ∈ R q [[ α, β ]; ( b j ) κ − j =0 ]. If κ ≥
2, then F c ∈ R q [[ α, β ]; ( c j ) κ − j =0 ].
6. A Schur–Nevanlinna type algorithm in the class R q ( C \ [ α, β ]) On the background of Proposition 3.7 and Theorem 3.45, we parametrized in [28, Sec. 8] the set M < q ([ α, β ]) of non-negative Hermitian q × q measures on [ α, β ] and generalized several resultsfrom the scalar theory of canonical moments (cf. [12]) to the matrix case. To that end, weassociated to such a measure the sequences built via Definitions 3.21 and 3.42 from its sequenceof power moments: Definition 6.1.
Let σ ∈ M < q ([ α, β ]) with sequence of power moments ( s ( σ ) j ) ∞ j =0 . Denoteby ( e ( σ ) j ) ∞ j =0 the [ α, β ]-interval parameter sequence of ( s ( σ ) j ) ∞ j =0 and by ( d ( σ ) j ) ∞ j =0 the sequenceof [ α, β ]-interval lengths associated with ( s ( σ ) j ) ∞ j =0 . Then we call ( e ( σ ) j ) ∞ j =0 the sequence ofmatricial canonical moments associated with σ and we say that ( d ( σ ) j ) ∞ j =0 is the sequence ofmatricial interval lengths associated with σ . Theorem 6.2 ( [28, Thm. 8.2]) . The mapping Π [ α,β ] : M < q ([ α, β ]) → E < q, ∞ ,δ given by σ ( e ( σ ) j ) ∞ j =0 is well defined and bijective. On the basis of Proposition 5.5, a parametrization of the class R q ( C \ [ α, β ]) immediatelyfollows: Definition 6.3.
Let F ∈ R q ( C \ [ α, β ]) with R [ α, β ]-measure ¨ σ F . Denote by ( e [ F ] j ) ∞ j =0 the se-quence of matricial canonical moments associated with ¨ σ F and by ( d [ F ] j ) ∞ j =0 the sequence of ma-tricial interval lengths associated with ¨ σ F . Then we call ( e [ F ] j ) ∞ j =0 the sequence of R [ α, β ] -Schur arameters associated with F and we say that ( d [ F ] j ) ∞ j =0 is the sequence of R [ α, β ] -intervallengths associated with F . Theorem 6.4.
The mapping ∆ [ α,β ] : R q ( C \ [ α, β ]) → E < q, ∞ ,δ given by F ( e [ F ] j ) ∞ j =0 is welldefined and bijective.Proof. Use Proposition 5.5 and Theorem 6.2.By means of this one-to-one correspondence, results obtained in [28, 29] on matricial canoni-cal moments associated with non-negative Hermitian measures from M < q ([ α, β ]) carry over tomatrix-valued functions belonging to the class R q ( C \ [ α, β ]) and their R [ α, β ]-Schur parame-ters.By virtue of Propositions 3.6 and 3.58, the F α,β -transformation (see Definitions 3.51and 3.55) for [ α, β ]-non-negative definite sequences of matrices gave rise to a transformationconsidered in [29, Def. 10.1] for non-negative Hermitian measures from M < q ([ α, β ]): Definition 6.5.
Let σ ∈ M < q ([ α, β ]) with sequence of power moments ( s j ) ∞ j =0 and let k ∈ N .Denote by ( s { k } j ) ∞ j =0 the k -th F α,β -transform of ( s j ) ∞ j =0 and by σ { k } the uniquely determinedelement in M < q, ∞ [[ α, β ]; ( s { k } j ) ∞ j =0 , =]. Then we call σ { k } the k -th M [ α, β ] -transform of σ . Remark . Let σ ∈ M < q ([ α, β ]). According to Definitions 6.5 and 3.55, then σ { } = σ and σ { k } is exactly the first M [ α, β ]-transform of σ { k − } for each k ∈ N . Remark . Let ( s j ) κj =0 ∈ F < q,κ,α,β , let σ ∈ M < q,κ [[ α, β ]; ( s j ) κj =0 , =], and let k ∈ Z ,κ .Then, in view of Definition 6.5 and Remark 3.57, it is readily checked that σ { k } belongsto M < q,κ − k [[ α, β ]; ( s { k } j ) κ − kj =0 , =].In view of Proposition 5.5, we can define a corresponding transformation for functions be-longing to the class R q ( C \ [ α, β ]): Definition 6.8.
Let F ∈ R q ( C \ [ α, β ]) with R [ α, β ]-measure σ and let k ∈ N . Denote by σ { k } the k -th M [ α, β ]-transform of σ and by F { k } the [ α, β ]-Stieltjes transform of σ { k } . Thenwe call F { k } the k -th R [ α, β ] -Schur transform of F . Remark . In the situation of Definition 6.8 we see from Proposition 5.5 that F { k } belongsto R q ( C \ [ α, β ]) and that σ { k } is the R [ α, β ]-measure of F { k } . Remark . Let F ∈ R q ( C \ [ α, β ]). Then, regarding Definition 6.8, Remark 6.6, and Propo-sition 5.5, it is readily checked that F { } = F and that F { k } is exactly the first R [ α, β ]-Schurtransform of F { k − } for each k ∈ N . Remark . Let ( s j ) κj =0 ∈ F < q,κ,α,β , let F ∈ R q [[ α, β ]; ( s j ) κj =0 ], and let k ∈ Z ,κ . Because ofRemarks 6.7 and 6.9, then F { k } ∈ R q [[ α, β ]; ( s { k } j ) κ − kj =0 ].One of the results in [29] states that the M [ α, β ]-transformation of a non-negative Hermitianmeasure from M < q ([ α, β ]) is essentially equivalent to left shifting its sequence of matricialcanonical moments: Proposition 6.12 ( [29, Prop. 10.4]) . Let k ∈ N and let σ ∈ M < q ([ α, β ]) with k -th M [ α, β ] -transform µ . Then e ( µ )0 = δ k − d ( σ ) k and e ( µ ) j = e ( σ ) k + j for all j ∈ N . Furthermore, d ( µ ) j = δ k d ( σ ) k + j for all j ∈ N . In particular, µ ([ α, β ]) = δ k − d ( σ ) k . R q ( C \ [ α, β ]) justi-fies the notions R [ α, β ] -Schur parameters and R [ α, β ] -Schur transform chosen in Definitions 6.3and 6.8, resp. Proposition 6.13.
Let k ∈ N and let F ∈ R q ( C \ [ α, β ]) with k -th R [ α, β ] -Schur transform G . Then e [ G ]0 = δ k − d [ F ] k and e [ G ] j = e [ F ] k + j for all j ∈ N . Furthermore, d [ G ] j = δ k d [ F ] k + j forall j ∈ N .Proof. In view of Definitions 6.3 and 6.8 and Remark 6.9, this is an immediate consequence ofProposition 6.12.Let Ω ∈ B R \ {∅} . A non-negative Hermitian measure σ ∈ M < q (Ω) is said to be molecular if there exists a finite subset B of Ω satisfying σ (Ω \ B ) = O q × q . Obviously, this is equivalentto the existence of an m ∈ N and sequences ( ξ ℓ ) mℓ =1 and ( A ℓ ) mℓ =1 from Ω and C q × q < , resp., suchthat σ = P mℓ =1 δ ξ ℓ A ℓ , where δ ξ ℓ is the Dirac measure on ([ α, β ] , B [ α,β ] ) with unit mass at ξ ℓ .It was shown in [29, Prop. 10.5] that σ ∈ M < q ([ α, β ]) is molecular if and only if for some k ∈ N its k -th M [ α, β ]-transform σ { k } coincides with the q × q zero measure on ([ α, β ] , B [ α,β ] ).This leads to a characterization of rational matrix-valued functions from R q ( C \ [ α, β ]) in termsof their k -th R [ α, β ]-Schur transforms: Proposition 6.14.
Let F ∈ R q ( C \ [ α, β ]) . Then the following statements are equivalent:(i) There exist complex q × q matrix polynomials P and Q such that det Q does not vanishidentically and that F coincides with the restriction of P Q − onto C \ [ α, β ] .(ii) There exists an integer k ∈ N such that F { k } coincides with the constant functionon C \ [ α, β ] with value O q × q .Proof. First observe that the R [ α, β ]-measure σ := ¨ σ F given via Definition 4.10 belongs to M < q ([ α, β ]). According to Lemma 4.12, the restriction f of F onto Π + belongs to R ,q (Π + ).Furthermore, the spectral measure µ := σ f of f given via Definition 4.4 belongs to M < q ( R )and, in view of Lemma 4.12, fulfills µ ( R \ [ α, β ]) = O q × q and µ ( B ) = σ ( B ) for all B ∈ B [ α,β ] .In particular, µ is molecular if and only if σ is molecular. From [29, Prop. 10.5] we see that σ is molecular if and only if, for some k ∈ N , the k -th M [ α, β ]-transform σ { k } of σ given viaDefinition 6.5 coincides with the q × q zero measure on ([ α, β ] , B [ α,β ] ). For an arbitrary k ∈ N ,by virtue of Definitions 6.8 and 5.3 and Proposition 5.5, we infer that σ { k } is the q × q zeromeasure on ([ α, β ] , B [ α,β ] ) if and only if F { k } is the constant function on C \ [ α, β ] with value O q × q . Consequently, we have shown that (ii) is equivalent to the following statement:(iii) µ is molecular.(i) ⇒ (iii): Suppose there exist complex q × q matrix polynomials P and Q such that det Q does not vanish identically and F is the restriction of P Q − onto C \ [ α, β ]. Then f coincideswith the restriction of P Q − onto Π + . Hence, the application of [17, Lem. B.4] yields (iii).(iii) ⇒ (i): Suppose that µ is molecular. From Proposition 5.5 we see that F is exactlythe [ α, β ]-Stieltjes transform ¨ S σ of σ given via Definition 5.3. If µ is the q × q zero measureon ( R , B R ), then σ coincides with the q × q zero measure on ([ α, β ] , B [ α,β ] ), hence ¨ S σ is, by(5.1), the constant function on C \ [ α, β ] with value O q × q , and, regarding F = ¨ S σ , thus (i)obviously holds true. Now assume that µ is not the q × q zero measure on ( R , B R ), i. e. µ ( R ) = O q × q . Proposition 5.4 shows that the moments s j := R [ α,β ] x j σ (d x ) exist for all j ∈ N z ∈ C with | z | > max {| α | , | β |} , furthermore z ∈ C \ [ α, β ] and¨ S σ ( z ) = − P ∞ j =0 z − ( j +1) s j . For all j ∈ N , obviously s j = R R x j µ (d x ). In view of (iii) and µ ( R ) = O q × q , then [17, Rem. 4.6] shows that ( s j ) ∞ j =0 is, using the terminology of [17], acompletely degenerate Hankel non-negative definite sequence of order n for some n ∈ N . Thus,from [17, Prop. 9.2 and Rem. 3.5] we obtain the existence of a constant ρ ∈ [0 , ∞ ) and specificcomplex q × q matrix polynomials a n and b n such that det b n does not vanish identically and P ∞ j =0 z − ( j +1) s j = ( a n b − n )( z ) holds true for all z ∈ C with | z | > ρ . Setting P := − a n and Q := b n , then det Q does not vanish identically and F ( z ) = ¨ S σ ( z ) = − ∞ X j =0 z − ( j +1) s j = − ( a n b − n )( z ) = ( P Q − )( z )is valid for all z ∈ C with | z | > max {| α | , | β | , ρ } . Since F is holomorphic in C \ [ α, β ], then polesof the matrix-valued rational function P Q − can only occur in [ α, β ] and F coincides with therestriction of P Q − onto C \ [ α, β ]. Consequently, (i) is valid.
7. The class
P R q ( C \ [ α, β ]) In the scalar case q = 1, the set of all solutions of problem FP [[ α, β ]; ( s j ) n +1 j =0 ] can beparametrized with functions of the class R ( C \ [ α, β ]) augmented by the constant functionwith value ∞ defined on C \ [ α, β ] (cf. [42, Thm. 7.2]). The corresponding approach forthe matricial situation q ≥ R q ( C \ [ α, β ]) of holomorphicmatrix-valued functions according to Appendix D to some class of regular q × q matrix pairsof meromorphic matrix-valued functions. Such a class was already considered in [10, Sec. 5].As a first step, we extend the class R q (Π + ), using the terminology from Appendix D and theend of Appendix F, without explaining these notations here. We only recall that a p × q matrixpair [ P ; Q ] is said to be regular if it satisfies rank (cid:2) PQ (cid:3) = q . Furthermore, we observe that theset P ( F ) of poles of any meromorphic matrix-valued function F is discrete. Notation . Denote by PR q (Π + ) the set of all ordered pairs [ P ; Q ] consisting of C q × q -valuedfunctions P and Q which are meromorphic in Π + , such that a discrete subset D of Π + exists,satisfying the following three conditions:(I) P ( P ) ∪ P ( Q ) ⊆ D .(II) rank h P ( z ) Q ( z ) i = q for all z ∈ Π + \ D .(III) Im([ Q ( z )] ∗ [ P ( z )]) ∈ C q × q < for all z ∈ Π + \ D .Using a continuity argument, the following result is readily checked: Remark . If [ P ; Q ] ∈ PR q (Π + ), then Im([ Q ( z )] ∗ [ P ( z )]) ∈ C q × q < for all z ∈ Π + \ [ P ( P ) ∪P ( Q )].Now we supplement Notation 7.1 in the following way: Notation . For each [ P ; Q ] ∈ PR q (Π + ), denote by E ([ P ; Q ]) the set of all z ∈ Π + \ [ P ( P ) ∪P ( Q )] satisfying rank h P ( z ) Q ( z ) i = q .Regarding Definition D.1, for each [ P ; Q ] ∈ PR q (Π + ), we see that E ([ P ; Q ]) is exactly the setof all points z ∈ Π + at which P and Q are both defined and the q × q matrix pair [ P ( z ); Q ( z )]is not regular. In general, the linear subspace R ( (cid:2) P ( z ) Q ( z ) (cid:3) ) depends on z , whereas its dimension26s well as the linear subspaces R ( Q ( z )), R ( P ( z )), Q ( z )( N ( P ( z ))), P ( z )( N ( Q ( z ))), and thedifference dim R ( P ( z )) − dim( P ( z )( N ( Q ( z )))) are essentially independent of z : Proposition 7.4.
Let [ P ; Q ] ∈ PR q (Π + ) . Then P := P ( P ) ∪ P ( Q ) is a discrete subsetof Π + and E := E ([ P ; Q ]) is a discrete subset of G := Π + \ P admitting the representation E = { z ∈ G : det[ Q ( z ) − i P ( z )] = 0 } . The set A := P ∪ E is the smallest discrete subsetof Π + satisfying the conditions (I)–(III) in Notation 7.1. For all z ∈ Π + \ A , the q × q matrixpair [ P ( z ); Q ( z )] is regular. For all z, w ∈ Π + \ A , furthermore R ( Q ( z )) = R ( Q ( w )) , R ( P ( z )) = R ( P ( w )) ,Q ( z )( N ( P ( z ))) = Q ( w )( N ( P ( w ))) , P ( z )( N ( Q ( z ))) = P ( w )( N ( Q ( w ))) , and dim R ( P ( z )) − dim( P ( z )( N ( Q ( z )))) = dim R ( P ( w )) − dim( P ( w )( N ( Q ( w )))) hold true.Proof. Observe that the matrix-valued functions P and Q are both meromorphic in Π + . Hence,the sets P ( P ) and P ( Q ) of poles as well as their union P are discrete subsets of Π + . Consideran arbitrary discrete subset D of Π + , satisfying the conditions (I)–(III) in Notation 7.1. Sucha subset exists by virtue of Notation 7.1. In view of Notation 7.1(II), and Notation 7.3, thenthe set E is a subset of D and hence discrete. In particular, E is a discrete subset of G .Because of Notation 7.3 and Remark 7.2, the conditions (I)–(III) in Notation 7.1 are fulfilledwhere the set D is substituted by A . Due to Notation 7.1(I), we have P ⊆ D . Takingadditionally into account
E ⊆ D , we see that the set A is a subset of D and thus a discretesubset of Π + . Therefore, the set A is the smallest discrete subset D of Π + satisfying theconditions (I)–(III) in Notation 7.1. Obviously, the matrix-valued functions F := Q + i P and G := Q − i P are both holomorphic in G . From Lemma D.10 we infer that, for all z ∈ G with det G ( z ) = 0, the q × q matrix pair [ P ( z ); Q ( z )] is regular, implying z / ∈ E . In view ofNotation 7.3 and Definition D.1, we can conclude that, for all z ∈ G \ E , the q × q matrix pair[ P ( z ); Q ( z )] is regular and fulfills Im([ Q ( z )] ∗ [ P ( z )]) ∈ C q × q < , by virtue of Remark 7.2. Becauseof Lemma D.11, we conversely have then det G ( z ) = 0 for all z ∈ G with z / ∈ E . Consequently, E = { z ∈ G : det G ( z ) = 0 } . Hence, S : G \ E → C p × q defined by S ( z ) := [ F ( z )][ G ( z )] − is aholomorphic matrix-valued function. Furthermore, due to Lemma D.11, we have k S ( z ) k S ≤ z ∈ G \E . In particular, S and − S both belong to the class S q × q ( G \E ) of Schur functions(in
G \E ) introduced in Notation F.9. In view of Lemma F.10, thus R ( I q ± S ( z )) = R ( I q ± S ( w ))and N ( I q ± S ( z )) = N ( I q ± S ( w )) hold true for all z, w ∈ G \ E . Regarding additionallyΠ + \ A = G \ E , the application of Lemma D.10 completes the proof.After transition to an appropriate equivalence relation, we can identify the class R q (Π + )of matrix-valued functions with the set of equivalence classes of pairs [ P ; Q ] ∈ PR q (Π + ) forwhich det Q does not identically vanish in Π + . The analogous considerations are worked outin detail below for the following extension PR q ( C \ [ α, β ]) of the class R q ( C \ [ α, β ]) in theabove mentioned sense. Notation . Denote by PR q ( C \ [ α, β ]) the set of all ordered pairs [ P ; Q ]consisting of C q × q -valued functions P and Q which are meromorphic in C \ [ α, β ], for which adiscrete subset D of C \ [ α, β ] exists, satisfying the following conditions:(I) P ( P ) ∪ P ( Q ) ⊆ D .(II) rank h P ( z ) Q ( z ) i = q for all z ∈ C \ ([ α, β ] ∪ D ).27III) z Im(( z − α )[ Q ( z )] ∗ [ P ( z )]) ∈ C q × q < for all z ∈ C \ ( R ∪ D ).(IV) z Im(( β − z )[ Q ( z )] ∗ [ P ( z )]) ∈ C q × q < for all z ∈ C \ ( R ∪ D ).Again using a continuity argument, the following result is readily checked: Remark . If [ P ; Q ] ∈ PR q ( C \ [ α, β ]), then z Im(( z − α )[ Q ( z )] ∗ [ P ( z )]) ∈ C q × q < and z Im(( β − z )[ Q ( z )] ∗ [ P ( z )]) ∈ C q × q < for all z ∈ C \ [ R ∪ P ( P ) ∪ P ( Q )].As done for Notation 7.3 above, we analogously supplement Notation 7.5 in the followingway: Notation . For each [ P ; Q ] ∈ PR q ( C \ [ α, β ]) denote by ¨ E ([ P ; Q ]) the set of all z ∈ C \ ([ α, β ] ∪ P ( P ) ∪ P ( Q )) satisfying rank (cid:2) P ( z ) Q ( z ) (cid:3) = q .Regarding Definition D.1, we see that, for each [ P ; Q ] ∈ PR q ( C \ [ α, β ]), the set ¨ E ([ P ; Q ])is exactly the set of all points z ∈ C \ [ α, β ] at which P and Q are both defined and for whichthe q × q matrix pair [ P ( z ); Q ( z )] is not regular. The pairs belonging to PR q ( C \ [ α, β ]) fulfillconditions analogous to those in Proposition 4.14 for matrix-valued functions belonging to R q ( C \ [ α, β ]): Lemma 7.8.
Let [ P ; Q ] ∈ PR q ( C \ [ α, β ]) and let P := P ( P ) ∪ P ( Q ) . Then z Im([ Q ( z )] ∗ [ P ( z )]) ∈ C q × q < for all z ∈ C \ ( R ∪ P ) and Re([ Q ( w )] ∗ [ P ( w )]) ∈ C q × q < forall w ∈ [ H − ( α )] \ P and − Re([ Q ( w )] ∗ [ P ( w )]) ∈ C q × q < for all w ∈ [ H + ( β )] \ P . Furthermore, [ Q ( x )] ∗ [ P ( x )] ∈ C q × q < for all x ∈ ( −∞ , α ) \P and − [ Q ( x )] ∗ [ P ( x )] ∈ C q × q < for all x ∈ ( β, ∞ ) \P .Proof. Consider an arbitrary z ∈ C \ ( R ∪ P ). We haveIm(( β − z )[ Q ( z )] ∗ [ P ( z )]) + Im(( z − α )[ Q ( z )] ∗ [ P ( z )]) = ( β − α ) Im([ Q ( z )] ∗ [ P ( z )]) . Regarding α < β , we obtain, by virtue of Remarks 7.6 and A.24, consequently z Im([ Q ( z )] ∗ [ P ( z )]) ∈ C q × q < . Let A := [ Q ( z )] ∗ [ P ( z )]. Using Remarks A.2, A.24, and 7.6,we infer in the case Re z < α thenRe A = 1Im z Im( zA ) − Re z Im z Im A < z Im( zA ) − α Im z Im A = 1Im z Im[( z − α ) A ] < O q × q , i. e., Re([ Q ( z )] ∗ [ P ( z )]) ∈ C q × q < . In the case Re z > β , we can conclude analogously − Re A = Re z Im z Im( A ) − z Im( zA ) < β Im z Im( A ) − z Im( zA ) = 1Im z Im[( β − z ) A ] < O q × q , i. e., − Re([ Q ( z )] ∗ [ P ( z )]) ∈ C q × q < . Observe that the matrix-valued functions P and Q are bothmeromorphic in C \ [ α, β ]. Since the set P is the union of the poles of P and Q , it is a discretesubset of C \ [ α, β ]. Furthermore, P and Q are both holomorphic in C \ ([ α, β ] ∪P ). Consequently,a continuity argument shows that we have Re([ Q ( w )] ∗ [ P ( w )]) ∈ C q × q < for all w ∈ [ H − ( α )] \ P and − Re([ Q ( w )] ∗ [ P ( w )]) ∈ C q × q < for all w ∈ [ H + ( β )] \ P . Regarding the continuity of thefunction S : C \ ([ α, β ] ∪ P ) → C q × q defined by S ( z ) := [ Q ( z )] ∗ [ P ( z )], we can conclude as inthe proof of Lemma C.3 that Im([ Q ( x )] ∗ [ P ( x )]) = O q × q holds true for all x ∈ R \ ([ α, β ] ∪ P ).Therefore, we get [ Q ( x )] ∗ [ P ( x )] = Re([ Q ( x )] ∗ [ P ( x )]) for all x ∈ R \ ([ α, β ] ∪ P ). Takinginto account the already shown inequalities, we can infer then [ Q ( x )] ∗ [ P ( x )] ∈ C q × q < for all x ∈ ( −∞ , α ) \ P and − [ Q ( x )] ∗ [ P ( x )] ∈ C q × q < for all x ∈ ( β, ∞ ) \ P .28y virtue of Lemma 7.8, we can think of PR q ( C \ [ α, β ]) as a subclass of PR q (Π + ) viarestricting to the open upper half-plane Π + . Analogous as done for the class PR q (Π + ) inProposition 7.4 above, we are now going to prove that for each pair [ P ; Q ] ∈ PR q ( C \ [ α, β ])certain linear subspaces associated with the q × q matrix pair [ P ( z ); Q ( z )] are essentially inde-pendent of z . This is in accordance with Proposition 4.15. In the proof we will use Lemma 7.8to reduce the situation to several open half-planes, in order to apply Proposition 7.4. Proposition 7.9.
Let [ P ; Q ] ∈ PR q ( C \ [ α, β ]) . Let Π := Π + , Π := H − ( α ) , , Π := Π − ,and Π := H + ( β ) . Then P := P ( P ) ∪ P ( Q ) is a discrete subset of C \ [ α, β ] and E := ¨ E ([ P ; Q ]) is a discrete subset of G := C \ ([ α, β ] ∪ P ) with Π k ∩ E = { z ∈ Π k \ P : det[ Q ( z ) − i k P ( z )] = 0 } for each k ∈ { , , , } . The set A := P ∪ E is the smallest discrete subset D of C \ [ α, β ] satisfying the conditions (I)–(IV) in Notation 7.5. For all z ∈ C \ ([ α, β ] ∪ A ) , the q × q matrixpair [ P ( z ); Q ( z )] is regular. For every choice of z and w in C \ ([ α, β ] ∪ A ) , furthermore R ( Q ( z )) = R ( Q ( w )) , R ( P ( z )) = R ( P ( w )) , (7.1) Q ( z )( N ( P ( z ))) = Q ( w )( N ( P ( w ))) , P ( z )( N ( Q ( z ))) = P ( w )( N ( Q ( w ))) , (7.2) and dim R ( P ( z )) − dim( P ( z )( N ( Q ( z )))) = dim R ( P ( w )) − dim( P ( w )( N ( Q ( w )))) hold true.Proof. Since P and Q are matrix-valued functions meromorphic in C \ [ α, β ], the union P of their poles is a discrete subset of C \ [ α, β ]. Consider an arbitrary discrete subset D of C \ [ α, β ] satisfying the conditions (I)–(IV) in Notation 7.5. Such a subset exists by virtue ofNotation 7.5. In view of Notation 7.5(II) and Notation 7.7, then the set E is a subset of D and hence discrete. In particular, E is a discrete subset of G . Because of Notation 7.7 andRemark 7.6, the conditions (I)–(IV) in Notation 7.5 are fulfilled with the set A instead of D .Due to Notation 7.5(I), we have P ⊆ D . Taking additionally into account
E ⊆ D , we see thatthe set A is a subset of D and thus a discrete subset of C \ [ α, β ]. Therefore, the set A isthe smallest discrete subset D of C \ [ α, β ] satisfying the conditions (I)–(IV) in Notation 7.5.For all z ∈ C \ ([ α, β ] ∪ A ), we have furthermore z ∈ G and z / ∈ E , implying rank (cid:2) P ( z ) Q ( z ) (cid:3) = q according to Notation 7.7, which, in view of Definition D.1, shows that the q × q matrix pair[ P ( z ); Q ( z )] is regular. Let φ ( ω ) := ω , φ ( ω ) := i ω + α , φ ( ω ) := − ω , and φ ( ω ) := − i ω + β .It is readily checked that, for each k ∈ { , , , } , the mapping φ k : Π + → Π k is bijective andthat the union of their images Π , Π , Π , Π is exactly the whole domain C \ [ α, β ].Consider now an arbitrary k ∈ { , , , } . Since the inverse ψ k := φ − k of φ k is an affinebijection from Π k onto Π + and since the sets P and E are discrete, we can infer that D k := ψ k (Π k ∩ A ) fulfills D k = ψ k (Π k ∩ ( P ∪ E )) = ψ k (Π k ∩ P ) ∪ ψ k (Π k ∩ E ) and is a discrete subsetof Π + . Regarding Remark A.2 and Lemma 7.8, it is then readily checked that via P k ( ω ) := i k − P ( φ k ( ω )) and Q k ( ω ) := Q ( φ k ( ω )) (7.3)matrix-valued functions P k : ψ k (Π k \ P ( P )) → C q × q and Q k : ψ k (Π k \ P ( Q )) → C q × q are given,such that the pair [ P k ; Q k ] consists of C q × q -valued functions, which are meromorphic in Π + ,for which P k := P ( P k ) ∪ P ( Q k ) fulfills P k = ψ k (Π k ∩ P ( P )) ∪ ψ k (Π k ∩ P ( Q )) = ψ k (Π k ∩ P ) ⊆D k , and for which rank (cid:2) P k ( ω ) Q k ( ω ) (cid:3) = q and Im([ Q k ( ω )] ∗ [ P k ( ω )]) ∈ C q × q < hold true for all ω in ψ k (Π k \ A ) = ψ k (Π k \ (Π k ∩ A )) = Π + \ D k . Consequently, [ P k ; Q k ] ∈ PR q (Π + ). In viewof Proposition 7.4, then P k we see that is a discrete subset of Π + and E k := E ([ P k ; Q k ]) is adiscrete subset of G k := Π + \ P k , admitting the representation E k = { ω ∈ G k : det[ Q k ( ω ) − i P k ( ω )] = 0 } = n ω ∈ G k : det h Q ( φ k ( ω )) − i k P ( φ k ( ω )) i = 0 o . A k := P k ∪ E k is a discrete subset of Π + and the linear subspaces R ( Q k ( ω )), R ( P k ( ω )), Q k ( ω )( N ( P k ( ω ))), P k ( ω )( N ( Q k ( ω ))) and the difference of dimensionsdim R ( P k ( ω )) − dim( P k ( ω )( N ( Q k ( ω )))) are independent of ω ∈ Π + \ A k . Since φ k is an affinebijection from Π + onto Π k , we can conclude that Q k := Π k ∩ P fulfills Q k = φ k ( P k ) and is adiscrete subset of Π k and that F k := φ k ( E k ) is a discrete subset of H k := Π k \ Q k . We have H k = φ k (Π + \ P k ) = φ k ( G k ) and F k = n ζ ∈ H k : det h Q ( ζ ) − i k P ( ζ ) i = 0 o . (7.4)Moreover, B k := Q k ∪ F k fulfills B k = φ k ( P k ) ∪ φ k ( E k ) = φ k ( P k ∪ E k ) = φ k ( A k ) and is a discretesubset of Π k . Thus, Π k \ B k = φ k (Π + \ A k ). Furthermore, R ( Q ( ζ )), R ( P ( ζ )), Q ( ζ )( N ( P ( ζ ))), P ( ζ )( N ( Q ( ζ ))), and dim R ( P ( ζ )) − dim( P ( ζ )( N ( Q ( ζ )))) are, in view of (7.3), independent of ζ ∈ Π k \ B k , i. e., independent of ζ ∈ φ k (Π + \ A k ).We are now going to verify Π k ∩ E = F k . First consider an arbitrary ζ ∈ Π k ∩ E . Because of H k = Π k \ Q k = Π k \ (Π k ∩ P ) = Π k \ P (7.5)and Notation 7.7, we have ζ ∈ H k and rank (cid:2) P ( ζ ) Q ( ζ ) (cid:3) = q . To the contrary, assume ζ / ∈ F k . Inview of (7.4), then det[ Q ( ζ ) − i k P ( ζ )] = 0. By virtue of Lemma D.10, hence the q × q matrixpair [ P ( ζ ); Q ( ζ )] is regular, i. e., rank (cid:2) P ( ζ ) Q ( ζ ) (cid:3) = q , according to Definition D.1. Since this is acontradiction, we necessarily have ζ ∈ F k . Conversely, consider an arbitrary ζ ∈ F k . Becauseof (7.4) and (7.5), then ζ ∈ H k = Π k \ P ⊆ C \ ([ α, β ] ∪ P ). To the contrary, assume ζ / ∈ E . In view of Notation 7.7, then rank (cid:2) P ( ζ ) Q ( ζ ) (cid:3) = q . According to Definition D.1, hence the q × q matrix pair [ P ( ζ ); Q ( ζ )] is regular. Regarding (7.3) and Remark D.5, for ω := ψ k ( ζ ), wehave ω ∈ ψ k ( H k ) = G k = Π + \ P k anddet([ P k ( ω )] ∗ [ P k ( ω )] + [ Q k ( ω )] ∗ [ Q k ( ω )]) = det([ P ( ζ )] ∗ [ P ( ζ )] + [ Q ( ζ )] ∗ [ Q ( ζ )]) = 0 . Consequently, due to Remark D.5, the q × q matrix pair [ P k ( ω ); Q k ( ω )] is regular, i. e.,rank (cid:2) P k ( ω ) Q k ( ω ) (cid:3) = q , according to Definition D.1. By virtue of Notation 7.3, we have then ω / ∈ E k , implying ζ / ∈ F k . Since this is a contradiction, we see that ζ necessarily be-longs to E and therefore to Π k ∩ E . In view of (7.4) and (7.5), we obtain the equationsΠ k ∩E = F k = { ζ ∈ Π k \P : det[ Q ( ζ ) − i k P ( ζ )] = 0 } . As already shown, for each k ∈ { , , , } the set B k is a discrete subset of Π k and the entities R ( Q ( ζ )), R ( P ( ζ )), Q ( ζ )( N ( P ( ζ ))), P ( ζ )( N ( Q ( ζ ))), and dim R ( P ( ζ )) − dim( P ( ζ )( N ( Q ( ζ )))) are independent of ζ ∈ Π k \ B k . Inparticular, B := B ∪ · · · ∪ B is a discrete subset of Π ∪ · · · ∪ Π = C \ [ α, β ]. Therefore, thesets (Π ∩ Π ) \ B , (Π ∩ Π ) \ B , and (Π ∩ Π ) \ B are non-empty. Consequently, we can inferthat the linear subspaces R ( Q ( ζ )), R ( P ( ζ )), Q ( ζ )( N ( P ( ζ ))), P ( ζ )( N ( Q ( ζ ))), and the differ-ence of dimensions dim R ( P ( ζ )) − dim( P ( ζ )( N ( Q ( ζ )))) are independent of ζ in C \ ([ α, β ] ∪ B ).Because of P = [ k =1 (Π k ∩ P ) = [ k =1 Q k and E = [ k =1 (Π k ∩ E ) = [ k =1 F k , we have furthermore A = P ∪ E = ( Q ∪ F ) ∪ · · · ∪ ( Q ∪ F ) = B ∪ · · · ∪ B = B . Thus, (7.1),(7.2), and dim R ( P ( z )) − dim( P ( z )( N ( Q ( z )))) = dim R ( P ( w )) − dim( P ( w )( N ( Q ( w )))) followfor all z, w ∈ C \ ([ α, β ] ∪ A ). 30s is easily seen, the class PR q ( C \ [ α, β ]) is closed under right multiplication by meromorphicmatrix-valued functions R with not identically vanishing determinant: Remark . Let [ P ; Q ] ∈ PR q ( C \ [ α, β ]) and let R be a C q × q -valued function meromorphicin C \ [ α, β ] such that det R does not vanish identically in C \ [ α, β ]. Then [ P R ; QR ] ∈PR q ( C \ [ α, β ]).In view of the remarks on meromorphic matrix-valued functions given at the end of Ap-pendix F, it is readily checked that an equivalence relation on the set PR q ( C \ [ α, β ]) is given.Regarding Remark D.8, this equivalence relation is in accordance with that considered inAppendix D for arbitrary p × q matrix pairs. Definition 7.11.
Two pairs [ P ; Q ] , [ S ; T ] ∈ PR q ( C \ [ α, β ]) are said to be equivalent if thereexists a C q × q -valued function R meromorphic in C \ [ α, β ] such that det R does not vanishidentically in C \ [ α, β ] which fulfills S = P R and T = QR . In this case, we write [ P ; Q ] ∼ [ S ; T ].Furthermore, denote by h [ P ; Q ] i the equivalence class of a pair [ P ; Q ] ∈ PR q ( C \ [ α, β ]) andby hQi := {h [ S ; T ] i : [ S ; T ] ∈ Q} the set of equivalence classes of pairs belonging to a subset Q of PR q ( C \ [ α, β ]).Using Remark 4.8, the following remark can be easily concluded from Proposition 4.18: Remark . Let F ∈ R q ( C \ [ α, β ]) and let the functions P, Q : C \ [ α, β ] → C q × q be defined by P ( z ) := F ( z ) and Q ( z ) := I q . Then the pair [ P ; Q ] belongs to PR q ( C \ [ α, β ])and det Q ( z ) = 0 holds true for all z ∈ C \ [ α, β ].Conversely, we have: Lemma 7.13 (cf. [10, Prop. 5.7]) . Let [ P ; Q ] ∈ PR q ( C \ [ α, β ]) be such that det Q does notidentically vanish in C \ [ α, β ] . Then F := P Q − belongs to R q ( C \ [ α, β ]) . Furthermore,the pair [ S ; T ] consisting of the functions S, T : C \ [ α, β ] → C q × q defined by S ( z ) := F ( z ) and T ( z ) := I q belongs to PR q ( C \ [ α, β ]) and fulfills [ P ; Q ] ∼ [ S ; T ] and det T ( z ) = 0 forall z ∈ C \ [ α, β ] .Proof. Due to [10, Prop. 5.7], we have F ∈ R q ( C \ [ α, β ]). In view of Remark 7.12, we getthen [ S ; T ] ∈ PR q ( C \ [ α, β ]) and det T ( z ) = 0 for all z ∈ C \ [ α, β ]. Furthermore, R := Q − is a C q × q -valued function, which is meromorphic in C \ [ α, β ], satisfying S = P R and T = QR .Since det R = (det Q ) − does not identically vanish in C \ [ α, β ], thus [ P ; Q ] ∼ [ S ; T ] follows.We end this section with an example of a simple family of pairs belonging to PR q ( C \ [ α, β ]). Remark . Let z ∈ C , let x := z − α , and let y := β − z . Involving δ := β − α , it is readilychecked that i( yx − xy ) = 2 δ Im( z ) and | y | x + | x | y = δyx .Given two complex matrices A and B , we will use the notation A ⊕ B := " A OO B . (7.6) Example 7.15.
Let
X, Y ∈ C q × q satisfy rank (cid:2) XY (cid:3) = q and Y ∗ X ∈ C q × q < . Let P, Q : C \ [ α, β ] → C q × q be defined by P ( z ) := X and Q ( z ) := Y and let g, h : C \ [ α, β ] → C be given by g ( z ) := z − α and h ( z ) := β − z , resp. Denote by I q and O q the constant q × q matrix-valuedfunctions defined on C \ [ α, β ] with values I q and O q × q , resp. Then:(a) The pairs [ P ; hQ ] and [ − P ; gQ ] belong to PR q ( C \ [ α, β ]).31b) If X = O q × q , then [ P ; hQ ] and [ − P ; gQ ] are equivalent to [ O q ; I q ].(c) If Y = O q × q , then [ P ; hQ ] and [ − P ; gQ ] are equivalent to [ I q ; O q ].We verify that statements (a)–(c) are true:(a) The functions P , hQ , − P , and gQ are holomorphic in C \ [ α, β ]. Consider an arbitrary z ∈ C \ [ α, β ]. Let x := z − α and let y := β − z . Observe that the matrices I q ⊕ ( yI q ) and ( − I q ) ⊕ ( xI q )are invertible. Consequently, we get rank (cid:2) P ( z ) h ( z ) Q ( z ) (cid:3) = rank (cid:2) XyY (cid:3) = rank([ I q ⊕ ( yI q )] (cid:2) XY (cid:3) ) =rank (cid:2) XY (cid:3) = q and rank (cid:2) − P ( z ) g ( z ) Q ( z ) (cid:3) = rank (cid:2) − XxY (cid:3) = rank([( − I q ) ⊕ ( xI q )] (cid:2) XY (cid:3) ) = rank (cid:2) XY (cid:3) = q .From the first equation in Remark 7.14 we can conclude Im( xy ) = δ Im z and Im( yx ) = − δ Im z .Taking additionally into account Y ∗ X ∈ C q × q < , we thus obtainIm(( z − α )[ h ( z ) Q ( z )] ∗ [ P ( z )]) = Im( xyY ∗ X ) = Im( xy ) Y ∗ X = δ Im( z ) Y ∗ X, Im(( β − z )[ h ( z ) Q ( z )] ∗ [ P ( z )]) = Im (cid:16) | y | Y ∗ X (cid:17) = O q × q , Im(( z − α )[ g ( z ) Q ( z )] ∗ [ − P ( z )]) = Im (cid:16) −| x | Y ∗ X (cid:17) = O q × q , and Im(( β − z )[ g ( z ) Q ( z )] ∗ [ − P ( z )]) = Im( − yxY ∗ X ) = − Im( yx ) Y ∗ X = δ Im( z ) Y ∗ X. Now assume in addition z / ∈ R . Because of δ > Y ∗ X ∈ C q × q < , we can infer1Im z Im(( z − α )[ h ( z ) Q ( z )] ∗ [ P ( z )]) ∈ C q × q < , z Im(( β − z )[ h ( z ) Q ( z )] ∗ [ P ( z )]) ∈ C q × q < , and1Im z Im(( z − α )[ g ( z ) Q ( z )] ∗ [ − P ( z )]) ∈ C q × q < , z Im(( β − z )[ g ( z ) Q ( z )] ∗ [ − P ( z )]) ∈ C q × q < , by virtue of Remark A.24. Hence, [ P ; hQ ] and [ − P ; gQ ] belong to PR q ( C \ [ α, β ]).(b) Assume X = O q × q . Then [ P ; hQ ] = [ O q ; hQ ] and [ − P ; gQ ] = [ O q ; gQ ]. Sincerank (cid:2) O q × q Y (cid:3) = rank (cid:2) XY (cid:3) = q , we have det Y = 0. Hence, det( hQ ) and det( gQ ) both do notvanish identically in C \ [ α, β ]. Because of " P ( hQ ) − ( hQ )( hQ ) − = " O q ( hQ ) − ( hQ )( hQ ) − = " O q I q and " ( − P )( gQ ) − ( gQ )( gQ ) − = " O q ( gQ ) − ( gQ )( gQ ) − = " O q I q , the pairs [ P ; hQ ] and [ − P ; gQ ] are then both equivalent to [ O q ; I q ].(c) Assume Y = O q × q . Then [ P ; hQ ] = [ P ; O q ] and [ − P ; gQ ] = [ − P ; O q ]. Sincerank (cid:2) XO q × q (cid:3) = rank (cid:2) XY (cid:3) = q , we have det X = 0. Hence, det P and det( − P ) both do notvanish identically in C \ [ α, β ]. Because of " P P − ( hQ ) P − = " P P − O q P − = " I q O q and " ( − P )( − P ) − ( gQ )( − P ) − = " ( − P )( − P ) − O q ( − P ) − = " I q O q , the pairs [ P ; hQ ] and [ − P ; gQ ] are then both equivalent to [ I q ; O q ].32 . The class of parameters The pairs belonging to the subclass of PR q ( C \ [ α, β ]) introduced below generate the equivalenceclasses, which will be used in Section 14 as parameters in the description of the set of allsolutions to Problem FP [[ α, β ]; ( s j ) mj =0 ]. Notation . For each M ∈ C q × p , let ¨ P [ M ] be the set of all pairs [ F ; G ] ∈ PR q ( C \ [ α, β ]) forwhich there exists a z ∈ C \ ([ α, β ] ∪ P ( F ) ∪ P ( G ) ∪ ¨ E ([ F ; G ])) such that R ( F ( z )) ⊆ R ( M ). Remark . If M ∈ C q × p fulfills rank M = q , then ¨ P [ M ] = PR q ( C \ [ α, β ]).The class ¨ P [ M ] can be characterized by an additional equation involving the transformationmatrix P R ( M ) corresponding to the orthogonal projection onto the column space R ( M ): Lemma 8.3.
Let M ∈ C q × p and let [ F ; G ] ∈ PR q ( C \ [ α, β ]) . Then [ F ; G ] ∈ ¨ P [ M ] if andonly if P R ( M ) F = F . In this case, R ( F ( z )) ⊆ R ( M ) for all z ∈ C \ ([ α, β ] ∪ P ( F )) .Proof. First observe that F is a matrix-valued function meromorphic in C \ [ α, β ] and that G := C \ ([ α, β ] ∪ P ( F )) is exactly the set of points at which F is holomorphic. Therefore, P R ( M ) F isa matrix-valued function meromorphic in C \ [ α, β ] and G is exactly the set of points at which P R ( M ) F is holomorphic. In view of Proposition 7.9, the set A := P ( F ) ∪ P ( G ) ∪ ¨ E ([ F ; G ]) is adiscrete subset of C \ [ α, β ] and R ( F ( z )) = R ( F ( w )) holds true for all z, w ∈ C \ ([ α, β ] ∪ A ).Assume [ F ; G ] ∈ ¨ P [ M ]. Then there exists some z ∈ C \ ([ α, β ] ∪ A ) with R ( F ( z )) ⊆ R ( M ).As a subset of A the set D := P ( G ) ∪ ¨ E ([ F ; G ]) is discrete and the function F which isholomorphic in G fulfills, for all w ∈ G \ D = C \ ([ α, β ] ∪ A ), furthermore R ( F ( w )) = R ( F ( z )) ⊆ R ( M ), implying P R ( M ) F ( w ) = F ( w ). Since D is discrete, the set G \ D has anaccumulation point in G . Therefore, the identity theorem for holomorphic functions yields P R ( M ) F ( z ) = F ( z ) for all z ∈ G , implying P R ( M ) F = F and R ( F ( z )) ⊆ R ( M ) for all z ∈ G .Conversely, assume P R ( M ) F = F . Since the set A is discrete, there exists some z ∈ C \ ([ α, β ] ∪A ). In particular, z ∈ G and, consequently, P R ( M ) F ( z ) = F ( z ). Thus R ( F ( z )) ⊆R ( M ), implying [ F ; G ] ∈ ¨ P [ M ].Using Lemma 8.3, it is readily checked that the equivalence relation on the set PR q ( C \ [ α, β ])introduced in Definition 7.11 is compatible with the here considered subclass ¨ P [ M ] in thefollowing sense: Remark . Let M ∈ C q × p and let [ F ; G ] ∈ ¨ P [ M ]. Then [ ˜ F ; ˜ G ] ∈ ¨ P [ M ] for all [ ˜ F ; ˜ G ] ∈h [ F ; G ] i .We can obtain a description of the set of equivalence classes h ¨ P [ M ] i , depending on the rank r of the matrix M in terms of equivalence classes of pairs belonging to PR r ( C \ [ α, β ]): Lemma 8.5.
Let M ∈ C q × p and let r := rank M :(a) If r = 0 , then h ¨ P [ M ] i = {h [ F ; G ] i} where F , G : C \ [ α, β ] → C q × q are defined by F ( z ) := O q × q and G ( z ) := I q .(b) Assume r ≥ . Let u , u , . . . , u r be an arbitrary orthonormal basis of R ( M ) , let U :=[ u , u , . . . , u r ] , and let Γ U : hPR r ( C \ [ α, β ]) i → h ¨ P [ M ] i be defined by Γ U ( h [ f ; g ] i ) := h [ U f U ∗ ; U gU ∗ + P [ R ( M )] ⊥ ] i . Then Γ U is well defined and bijective. roof. First assume r = 0, i. e., M = O q × p . Hence, P R ( M ) = O q × q . Consider now an arbitrarypair [ F ; G ] ∈ ¨ P [ M ]. By virtue of Lemma 8.3, then F = P R ( M ) F = F follows. Observe that[ F ; G ] belongs to PR q ( C \ [ α, β ]). In view of Notation 7.5, thus rank (cid:2) F ( z ) G ( z ) (cid:3) = q for some z ∈ C \ ([ α, β ] ∪P ( F ) ∪P ( G )). Because of F ( z ) = F ( z ) = O q × q , then necessarily det G ( z ) = 0holds true. In particular, det G does not vanish identically in C \ [ α, β ]. Consequently, theapplication of Lemma 7.13 to the pair [ F ; G ] and regarding F G − = F G − = F , we obtain[ F ; G ] ∼ [ F ; G ]. Therefore, h ¨ P [ M ] i ⊆ {h [ F ; G ] i} is verified. Using Remark 7.12 andLemma 8.3, we can easily infer [ F ; G ] ∈ ¨ P [ M ]. Hence, h ¨ P [ M ] i = {h [ F ; G ] i} .Now assume r ≥
1. We have U ∗ U = I r and R ( U ) = R ( M ). Let N := P [ R ( M )] ⊥ . UsingRemarks A.11 and A.12, we immediately obtain the representations N = P [ R ( U )] ⊥ = I q − P R ( U ) = I q − U U ∗ . (8.1)In particular, we see N U = U − U U ∗ U = O q × r and U ∗ N = U ∗ − U ∗ U U ∗ = O r × q . (8.2)We first consider an arbitrary pair [ f ; g ] ∈ PR r ( C \ [ α, β ]) and show that Γ U ( h [ f ; g ] i ) belongsto h ¨ P [ M ] i : According to Notation 7.5, the C r × r -valued functions f and g are meromorphic in C \ [ α, β ] and there exists a discrete subset D of C \ [ α, β ] with P ( f ) ∪ P ( g ) ⊆ D such thatrank (cid:2) f ( z ) g ( z ) (cid:3) = r for all z ∈ C \ ([ α, β ] ∪ D ) and furthermore z Im(( z − α )[ g ( z )] ∗ [ f ( z )]) ∈ C q × q < and z Im(( β − z )[ g ( z )] ∗ [ f ( z )]) ∈ C q × q < for all z ∈ C \ ( R ∪ D ) hold true. Obviously, F := U f U ∗ and G := U gU ∗ + P [ R ( M )] ⊥ (8.3)are C q × q -valued functions meromorphic in C \ [ α, β ] with P ( F ) ⊆ P ( f ) and P ( G ) ⊆ P ( g ).Thus, P ( F ) ∪ P ( G ) ⊆ D follows. Consider an arbitrary z ∈ C \ ([ α, β ] ∪ D ). In view ofDefinition D.1, the r × r matrix pair [ f ( z ); g ( z )] is regular. Furthermore, we have F ( z ) = U [ f ( z )] U ∗ and G ( z ) := U [ g ( z )] U ∗ + P [ R ( M )] ⊥ . By virtue of Lemma D.12, then [ F ( z ); G ( z )] isa regular q × q matrix pair fulfilling R ( F ( z )) ⊆ R ( U ) and [ G ( z )] ∗ [ F ( z )] = U ([ g ( z )] ∗ [ f ( z )]) U ∗ .According to Definition D.1, in particular rank (cid:2) F ( z ) G ( z ) (cid:3) = q . In the case z / ∈ R , using Remarks A.2and A.25, we can infer z Im(( z − α )[ G ( z )] ∗ [ F ( z )]) = U [ z Im(( z − α )[ g ( z )] ∗ [ f ( z )])] U ∗ ∈ C q × q < and similarly z Im(( β − z )[ G ( z )] ∗ [ F ( z )]) ∈ C q × q < . So D is a discrete subset of C \ [ α, β ] suchthat the conditions (I)–(IV) in Notation 7.5 are fulfilled for [ P ; Q ] = [ F ; G ]. Consequently,[ F ; G ] belongs to PR q ( C \ [ α, β ]). Because of rank (cid:2) F ( z ) G ( z ) (cid:3) = q , we see from Notation 7.7that z / ∈ ¨ E ([ F ; G ]). Summarizing, we have z ∈ C \ ([ α, β ] ∪ P ( F ) ∪ P ( G ) ∪ ¨ E ([ F ; G ])) and R ( F ( z )) ⊆ R ( U ) = R ( M ). Therefore, we obtain [ F ; G ] ∈ ¨ P [ M ] and, regarding (8.3), thusΓ U ( h [ f ; g ] i ) ∈ h ¨ P [ M ] i follows.Next we are going to show that Γ U ( h [ f ; g ] i ) is independent of the choice of the particularrepresentative [ f ; g ] of the equivalence class h [ f ; g ] i ∈ hPR r ( C \ [ α, β ]) i : To that end, considertwo arbitrary pairs [ f ; g ] and [ f ; g ] from PR r ( C \ [ α, β ]) satisfying [ f ; g ] ∼ [ f ; g ]. Foreach j ∈ { , } , let F j := U f j U ∗ and G j := U g j U ∗ + N. (8.4)According to Definition 7.11, there is a C r × r -valued function ρ meromorphic in C \ [ α, β ]such that det ρ does not vanish identically in C \ [ α, β ], for which f = f ρ and g = g ρ .34et R := U ρU ∗ + N . Then R is a C q × q -valued function meromorphic in C \ [ α, β ]. Regarding U ∗ U = I r , N = N , and (8.2), we get F R = U f U ∗ U ρU ∗ + U f U ∗ N = U f ρU ∗ = U f U ∗ = F and G R = U g U ∗ U ρU ∗ + U g U ∗ N + N U ρU ∗ + N = U g ρU ∗ + N = U g U ∗ + N = G . Furthermore, there exists some z ∈ C \ [ α, β ] such that ρ is holomorphic at z with det ρ ( z ) = 0.In addition, we are going to check now that det R ( z ) = 0. For this reason, consider an arbitraryvector v ∈ N ( R ( z )). Let w := U ∗ v . Then we have U ρ ( z ) w + N v = [
U ρ ( z ) U ∗ + N ] v = R ( z ) v = O q × , (8.5)implying U ∗ U ρ ( z ) w + U ∗ N v = O r × . In view of U ∗ U = I r and (8.2), thus ρ ( z ) w = O r × follows. Because of det ρ ( z ) = 0, then necessarily w = O r × holds true. Substituting thisinto (8.5), we get N v = O q × . Regarding (8.1), hence v = U U ∗ v = U w = O q × . Therefore,the linear subspace N ( R ( z )) is trivial, implying det R ( z ) = 0. In particular, det R doesnot vanish identically in C \ [ α, β ]. According to Definition 7.11, then [ F ; G ] ∼ [ F ; G ].Consequently, Γ U ( h [ f ; g ] i ) = Γ U ( h [ f ; g ] i ). Thus, the mapping Γ U is well defined. We arenow going to show that the mapping Γ U is injective. To that end, consider two arbitrary pairs[ f ; g ] and [ f ; g ] from PR r ( C \ [ α, β ]) satisfying Γ U ( h [ f ; g ] i ) = Γ U ( h [ f ; g ] i ). Let [ F ; G ]and [ F ; G ] be given via (8.4). Then [ F ; G ] ∼ [ F ; G ]. Hence, according to Definition 7.11,there exists a C q × q -valued function R meromorphic in C \ [ α, β ] such that det R does not vanishidentically in C \ [ α, β ], fulfilling F = F R and G = G R . Let ρ := U ∗ RU . Then ρ is a C r × r -valued function meromorphic in C \ [ α, β ]. Regarding U ∗ U = I r , (8.4), and (8.2), we have f ρ = U ∗ U f U ∗ RU = U ∗ F RU = U ∗ F U = U ∗ U f U ∗ U = f and g ρ = U ∗ U g U ∗ RU = U ∗ ( G − N ) RU = U ∗ G RU = U ∗ G U = U ∗ U g U ∗ U = g . Furthermore, there exists some z ∈ C \ [ α, β ] such that R is holomorphic at z with det R ( z ) =0. In addition, we now prove that det ρ ( z ) = 0. To do this, we consider an arbitrary vector w ∈ N ( ρ ( z )). Let v := U w . Then U ∗ R ( z ) v = U ∗ R ( z ) U w = ρ ( z ) w = O r × . Becauseof (8.4) and (8.2), we have N G j = N = N for each j ∈ { , } . In view of G = G R and(8.1), we hence get N = N R = R − U U ∗ R . From (8.2) we infer N v = N U w = O q × . Takingadditionally into account U ∗ R ( z ) v = O r × , we can then conclude R ( z ) v = O q × . Sincedet R ( z ) = 0 holds true, necessarily v = O q × follows. Regarding U ∗ U = I r , we thus obtain w = U ∗ v = O r × . Therefore, the linear subspace N ( ρ ( z )) is trivial, implying det ρ ( z ) = 0.In particular, det ρ does not vanish identically in C \ [ α, β ]. According to Definition 7.11,consequently [ f ; g ] ∼ [ f ; g ], i. e., h [ f ; g ] i = h [ f ; g ] i .We finish the proof by showing that the mapping Γ U is surjective. To that end, consider anarbitrary pair [ F ; G ] from ¨ P [ M ]. Then [ F ; G ] ∈ PR q ( C \ [ α, β ]). According to Notation 7.5,the C q × q -valued functions F and G are meromorphic in C \ [ α, β ]. Consequently, B := G − i F is a C q × q -valued function meromorphic in C \ [ α, β ]. Due to Proposition 7.9, theset A := P ( F ) ∪ P ( G ) ∪ ¨ E ([ F ; G ]) is a discrete subset of C \ [ α, β ]. Consider an arbitrary w ∈ Π + \ A . In view of Proposition 7.9, the q × q matrix pair [ F ( w ); G ( w )] is regular. UsingLemma 7.8 and Remark A.24, we get Im([ G ( w )] ∗ [ F ( w )]) ∈ C q × q < . We see from Lemma 8.3moreover R ( F ( w )) ⊆ R ( M ). Therefore, Proposition D.13 applies to the q × q matrix pair[ F ( w ); G ( w )] and we get det B ( w ) = 0. In particular, det B does not vanish identically in C \ [ α, β ]. Hence, R := B − is a C q × q -valued function meromorphic in C \ [ α, β ]. Consequently, f := U ∗ F RU and g := U ∗ G RU (8.6)35re C r × r -valued functions meromorphic in C \ [ α, β ]. In addition, f and g are both holomorphicat w with f ( w ) = U ∗ [ F ( w )][ B ( w )] − U and g ( w ) = U ∗ [ G ( w )][ B ( w )] − U . Thus, F := U f U ∗ and G := U gU ∗ + N (8.7)are C q × q -valued functions, which are meromorphic in C \ [ α, β ] and holomorphic at w with F ( w ) = U [ f ( w )] U ∗ and G ( w ) = U [ g ( w )] U ∗ + P [ R ( M )] ⊥ . Due to Proposition D.13,the q × q matrix pair [ F ( w ); G ( w )] is regular and satisfies F ( w ) = [ F ( w )][ B ( w )] − and G ( w ) = [ G ( w )][ B ( w )] − . Let H ( F ), H ( G ), H ( R ), H ( F ), and H ( G ) be the sets of com-plex numbers at which F , G , R , F , and G are holomorphic, resp. Taking into account thearbitrary choice of w ∈ Π + \ A , we can infer that Π + \ A is a subset of each of the sets H ( F ), H ( G ), H ( R ), H ( F ), and H ( G ). Since A is discrete, the set Π + \ A has in particular anaccumulation point in H ( F ) ∩ H ( G ) ∩ H ( R ) ∩ H ( F ) ∩ H ( G ). Using the identity theoremfor holomorphic functions, we thus can conclude F = F R and G = G R. (8.8)Observe that det R = (det B ) − does not vanish identically in C \ [ α, β ]. Consequently,Remark 7.10 yields [ F ; G ] ∈ PR q ( C \ [ α, β ]) and [ F ; G ] ∼ [ F ; G ]. We are now go-ing to show that [ f ; g ] belongs to PR r ( C \ [ α, β ]): Since det B does not vanish identi-cally in C \ [ α, β ], we obtain from the identity theorem for holomorphic functions that N := { ζ ∈ C \ ([ α, β ] ∪P (det B )) : det B ( ζ ) = 0 } is a discrete subset of C \ [ α, β ]. As already men-tioned, A is a discrete subset of C \ [ α, β ]. Therefore, D := A∪P ( R ) ∪P ( B ) ∪N is a discrete sub-set of C \ [ α, β ] as well. In view of (8.6), we have P ( f ) ⊆ P ( F ) ∪P ( R ) and P ( g ) ⊆ P ( G ) ∪P ( R ).Hence, P ( f ) ∪P ( g ) ⊆ D follows. Consider an arbitrary z ∈ C \ ([ α, β ] ∪D ). Let P := U [ f ( z )] U ∗ and Q := U [ g ( z )] U ∗ + P [ R ( U )] ⊥ . By virtue of (8.1), (8.7), and (8.8), we immediately see that P = F ( z ) = [ F ( z )][ B ( z )] − and Q = G ( z ) = [ G ( z )][ B ( z )] − . Observe that, due to Propo-sition 7.9, the q × q matrix pair [ F ( z ); G ( z )] is regular. Because of Remark D.6, thus the q × q matrix pair [ P ; Q ] is regular and Q ∗ P = [ B ( z )] −∗ ([ G ( z )] ∗ [ F ( z )])[ B ( z )] − holds true.Regarding U ∗ U = I r , we can now apply Lemma D.12 to the r × r matrix pair [ f ( z ); g ( z )] tosee that [ f ( z ); g ( z )] is regular and that Q ∗ P = U ([ g ( z )] ∗ [ f ( z )]) U ∗ is fulfilled. In particular, weobtain rank (cid:2) f ( z ) g ( z ) (cid:3) = r , according to Definition D.1, and furthermore[ g ( z )] ∗ [ f ( z )] = U ∗ U [ g ( z )] ∗ [ f ( z )] U ∗ U = U ∗ Q ∗ P U = U ∗ [ B ( z )] −∗ ([ G ( z )] ∗ [ F ( z )])[ B ( z )] − U = (cid:16) [ B ( z )] − U (cid:17) ∗ ([ G ( z )] ∗ [ F ( z )]) (cid:16) [ B ( z )] − U (cid:17) . In the case z / ∈ R , due to Proposition 7.9, we have z Im(( z − α )[ G ( z )] ∗ [ F ( z )]) ∈ C q × q < and z Im(( β − z )[ G ( z )] ∗ [ F ( z )]) ∈ C q × q < , implying, by virtue of Remarks A.2 and A.25, then z Im(( z − α )[ g ( z )] ∗ [ f ( z )]) = ([ B ( z )] − U ) ∗ [ z Im(( z − α )[ G ( z )] ∗ [ F ( z )])]([ B ( z )] − U ) ∈ C r × r < and, similarly, z Im(( β − z )[ g ( z )] ∗ [ f ( z )]) ∈ C r × r < . According to Notation 7.5, hence [ f ; g ]belongs to PR r ( C \ [ α, β ]). Applying Γ U to the equivalence class of [ f ; g ], we get with (8.7)and [ F ; G ] ∼ [ F ; G ] then Γ U ( h [ f ; g ] i ) = h [ F ; G ] i = h [ F ; G ] i . Example 8.6.
Let M ∈ C q × p and let F, G : C \ [ α, β ] → C q × q be defined by F ( z ) := O q × q and G ( z ) := I q . Then [ F ; G ] ∈ ¨ P [ M ]. 36 xample 8.7. Let M ∈ C q × q H and let f, g : C \ [ α, β ] → C be both holomorphic and notidentically vanishing. Let U be a linear subspace of R ( M ) and let P := P U . Let F, G : C \ [ α, β ] → C q × q be defined by F ( z ) := f ( z ) M P M and G ( z ) := g ( z )( I q − M † P M ). Then [ F ; G ]belongs to ¨ P [ M ]. Indeed, since f and g are both holomorphic and not identically vanishing, thematrix-valued functions F and G are holomorphic and in particular meromorphic in C \ [ α, β ]and D := { z ∈ C \ [ α, β ] : f ( z ) = 0 or g ( z ) = 0 } is a discrete subset of C \ [ α, β ]. Observethat M ∗ = M implies ( M † ) ∗ = M † and M † M = M M † , by virtue of Remarks A.14 and A.18.Consider an arbitrary z ∈ C \ ([ α, β ] ∪ D ). Let v ∈ N ( (cid:2) F ( z ) G ( z ) (cid:3) ), i. e., f ( z ) M P M v = O q × and g ( z )( I q − M † P M ) v = O q × . Since f ( z ) = 0 and g ( z ) = 0, hence M P M v = O q × and v = M † P M v . Thus, taking additionally into account (3.3), we can conclude v = M † P M v = M † M M † P M v = M † M † M P M v = O q × . This shows rank (cid:2) F ( z ) G ( z ) (cid:3) = q . Observe that U ⊆ R ( M )implies M M † P = P , by virtue of Remarks A.11 and A.20. Taking additionally into account P ∗ = P = P , we can furthermore conclude( I q − M † P M ) ∗ ( M P M ) = ( I q − M P M † ) M P M = M P M − M P M † M P M = M P M − M P M M † P M = M P M − M P M = O q × q . Consequently, [ G ( z )] ∗ [ F ( z )] = g ( z ) f ( z )( I q − M † P M ) ∗ ( M P M ) = O q × q . If z / ∈ R , thus thematrices z Im(( z − α )[ G ( z )] ∗ [ F ( z )]) and z Im(( β − z )[ G ( z )] ∗ [ F ( z )]) are both non-negativeHermitian. Hence, [ F ; G ] belongs to PR q ( C \ [ α, β ]). Since obviously P R ( M ) F = F , then[ F ; G ] ∈ ¨ P [ M ] follows by virtue of Lemma 8.3.
9. The F α,β ( M ) -transformation and its inverse Our next considerations are aimed at preparing the foundations for the desired function-theoretic Schur–Nevanlinna type algorithm. This algorithm consists of two different instances,because the first step differs from the remaining ones. In this section, we treat the algebraic for-malism for the first step. Doing this, we take into account as well the forward as the backwardform of the algorithm.We are now going to introduce a transformation of matrix-valued functions, which is inti-mately connected with the F α,β -transformation for sequences of complex matrices (see Defini-tion 3.51).In this section, for an arbitrarily given complex matrix E , we write P E := P R ( E ) and Q E := P N ( E ) for the transformation matrix corresponding to the orthogonal projection onto R ( E ) and N ( E ),resp. In view of Remarks A.11, A.10, and A.18, we have R ( P E ) = R ( E ) , N ( P E ) = N ( E ∗ ) , R ( Q E ) = N ( E ) , N ( Q E ) = R ( E ∗ ) ,P E = P E , P ∗ E = P E , Q E = Q E , Q ∗ E = Q E , (9.1)and P E = EE † , P E ∗ = E † E, Q E = I q − E † E, Q E ∗ = I p − EE † . (9.2)In the sequel, we will also use these identities without explicitly mentioning. Furthermore, weconsider here a complex matrix M , which in the context of the matricial Hausdorff moment37roblem will later be the non-negative Hermitian matrix s taken from a sequence ( s j ) κj =0 belonging to F < q,κ,α,β . Definition 9.1.
Let G be a non-empty subset of C , let F : G → C p × q be a matrix-valuedfunction, and let M be a complex p × q matrix. Then the pair [ G ; G ] built with the functions G and G defined on G by G ( z ) := ( β − z ) F ( z ) − M and G ( z ) := ( β − z ) h ( z − α ) M † F ( z ) + P M ∗ i + δQ M is called the F α,β ( M ) -transformed pair of F .In connection with the F α,β ( M )-transformation, we consider the following quadratic( p + q ) × ( p + q ) matrix polynomial: Notation . Let M ∈ C p × q . Then let ¨ W M : C → C ( p + q ) × ( p + q ) be defined by¨ W M ( z ) := − ( β − z ) P M M − ( β − z )( z − α ) M † − ( β − z ) P M ∗ − δQ M . In what follows, we will use the notation given via (7.6) to calculate, in view of Remark A.36,certain forms involving the signature matrix ˜ J q given by (A.1). For an arbitrarily given z ∈ C ,we will furthermore write abbreviatory x := z − α and y := β − z . Obviously, we have y + x = β − α = δ and αy + βx = αβ − αz + βz − βα = ( β − α ) z = δz (9.3)as well as αx + βy = αz − α + β − βz = ( β + α )( β − α ) − ( β − α ) z = δ ( β + α − z ) . (9.4) Lemma 9.3.
Let M ∈ C q × q H . Let z ∈ C , let x := z − α , and let y := β − z .(a) Let W := ¨ W M ( z ) . Then W ∗ ˜ J q W = − | y | Im( z ) M † i( yx + | y | ) P M − i( yx + | y | ) P M − z ) M . (9.5) (b) Let W := [( xI q ) ⊕ I q ] W and let W := [( yI q ) ⊕ I q ] W . Then W ∗ ˜ J q W = δ " O q × q i yxP M − i yxP M − z ) M = O q × q i( | y | x + | x | y ) P M − i( | y | x + | x | y ) P M − δ Im( z ) M (9.6) and W ∗ ˜ J q W = δ | y | " − z ) M † i P M − i P M O q × q . (9.7) (c) Let W := [( yxI q ) ⊕ I q ] W . Then W ∗ ˜ J q W = | y | − | x | Im( z ) M † i( yx + | x | ) P M − i( yx + | x | ) P M − z ) M . (9.8)38n view of (9.1), (9.2), and Remarks A.14 and 7.14, the proof of Lemma 9.3 is straightforward.We omit the details.In addition, we now rewrite the right-hand sides of the equations (9.5)–(9.8), using thesignature matrix ˜ J q : Proposition 9.4.
Let M ∈ C q × q H . Let z ∈ C , let x := z − α , and let y := β − z .(a) Let W := ¨ W M ( z ) . Then W ∗ ˜ J q W = [( yxP M ) ⊕ I q ] ∗ ˜ J q [( yxP M ) ⊕ I q ]+ | y | h ( P M ⊕ I q ) ∗ ˜ J q ( P M ⊕ I q ) − z )( M † ⊕ O q × q ) i − z )( O q × q ⊕ M ) . (b) Let W := [( xI q ) ⊕ I q ] W and let W := [( yI q ) ⊕ I q ] W . Then W ∗ ˜ J q W = δ (cid:16) [( yxP M ) ⊕ I q ] ∗ ˜ J q [( yxP M ) ⊕ I q ] − z )( O q × q ⊕ M ) (cid:17) = | y | [( xP M ) ⊕ I q ] ∗ ˜ J q [( xP M ) ⊕ I q ]+ | x | [( yP M ) ⊕ I q ] ∗ ˜ J q [( yP M ) ⊕ I q ] − δ Im( z )( O q × q ⊕ M ) and W ∗ ˜ J q W = δ | y | h ( P M ⊕ I q ) ∗ ˜ J q ( P M ⊕ I q ) − z )( M † ⊕ O q × q ) i . (c) Let W := [( yxI q ) ⊕ I q ] W . Then W ∗ ˜ J q W = | y | (cid:26) [( yxP M ) ⊕ I q ] ∗ ˜ J q [( yxP M ) ⊕ I q ]+ | x | h ( P M ⊕ I q ) ∗ ˜ J q ( P M ⊕ I q ) − z )( M † ⊕ O q × q ) i − z )( O q × q ⊕ M ) (cid:27) . Proof.
Taking into account P ∗ M = P M and (A.1), the asserted identities immediately followfrom Lemma 9.3.It will be clear from Lemmata 9.13 and 9.14 below that the following transformation for pairsof meromorphic matrix-valued functions is, under certain conditions, essentially the inversionof the F α,β ( M )-transformation. To define this inverse transformation, we use the terminologygiven at the end of Appendix F. Definition 9.5.
Let G be a domain. Let G be a C p × q -valued function meromorphic in G and let G be a C q × q -valued function meromorphic in G . Let M ∈ C p × q , let the functions g, h : G → C be defined by g ( z ) := z − α and h ( z ) := β − z, (9.9)resp., and let F := hP M G + M G and F := − hgM † G + hG . (9.10)Suppose that det F does not identically vanish in G . Then we call the C p × q -valued function F := F F − (, which is meromorphic in G ,) the inverse F α,β ( M ) -transform of [ G ; G ].39o the inverse F α,β ( M )-transform we can associate the following matrix polynomial: Notation . Let M ∈ C p × q . Then let ¨ V M : C → C ( p + q ) × ( p + q ) be defined by¨ V M ( z ) := ( β − z ) P M M − ( β − z )( z − α ) M † ( β − z ) I q . Remark . Let M ∈ C q × q and let z ∈ C . Then:(a) If M = O q × q , then ¨ V M ( z ) = h O q × q O q × q O q × q ( β − z ) I q i .(b) If M is invertible, then ¨ V M ( z ) = h ( β − z ) I q M − ( β − z )( z − α ) M − ( β − z ) I q i .Regarding y + x = δ and Q M = I q − P M ∗ , it is readily checked that the matrix polynomials¨ V M and ¨ W M are connected in the following way by the signature matrix j pq given in (A.1): Remark . If z ∈ C , then [ ¨ V M ( z )] j pq = − j pq [ ¨ W M ( z ) + ( z − α )( O p × p ⊕ Q M )].Consequently, a result analogous to Proposition 9.4 follows: Proposition 9.9.
Let M ∈ C q × q H . Let z ∈ C , let x := z − α , and let y := β − z .(a) Let V := ¨ V M ( z ) . Then V ∗ ˜ J q V = [( yxP M ) ⊕ I q ] ∗ ˜ J q [( yxP M ) ⊕ I q ]+ | y | h ( P M ⊕ I q ) ∗ ˜ J q ( P M ⊕ I q ) + 2 Im( z )( M † ⊕ O q × q ) i + 2 Im( z )( O q × q ⊕ M ) . (b) Let V := [( xI q ) ⊕ I q ] V and let V := [( yI q ) ⊕ I q ] V . Then V ∗ ˜ J q V = δ (cid:16) [( yxP M ) ⊕ I q ] ∗ ˜ J q [( yxP M ) ⊕ I q ] + 2 Im( z )( O q × q ⊕ M ) (cid:17) = | y | [( xP M ) ⊕ I q ] ∗ ˜ J q [( xP M ) ⊕ I q ]+ | x | [( yP M ) ⊕ I q ] ∗ ˜ J q [( yP M ) ⊕ I q ] + 2 δ Im( z )( O q × q ⊕ M ) and V ∗ ˜ J q V = δ | y | h ( P M ⊕ I q ) ∗ ˜ J q ( P M ⊕ I q ) + 2 Im( z )( M † ⊕ O q × q ) i . (c) Let V := [( yxI q ) ⊕ I q ] V . Then V ∗ ˜ J q V = | y | (cid:26) [( yxP M ) ⊕ I q ] ∗ ˜ J q [( yxP M ) ⊕ I q ]+ | x | h ( P M ⊕ I q ) ∗ ˜ J q ( P M ⊕ I q ) + 2 Im( z )( M † ⊕ O q × q ) i + 2 Im( z )( O q × q ⊕ M ) (cid:27) . Proof.
Consider an arbitrary ℓ ∈ { , , , } . Using the notation given in Proposition 9.4,we obtain, by virtue of Remarks A.37 and 9.8, then V ℓ j qq = − j qq [ W ℓ + x ( O q × q ⊕ Q M )] and˜ J q j qq = − j qq ˜ J q . Taking additionally into account j qq = I q and j ∗ qq = j qq and setting U ℓ := W ℓ + x ( O q × q ⊕ Q M ), we can conclude hence V ∗ ℓ ˜ J q V ℓ = ( − j qq U ℓ j qq ) ∗ ˜ J q ( − j qq U ℓ j qq ) = j qq U ∗ ℓ ( j qq ˜ J q j qq ) U ℓ j qq = − j qq ( U ∗ ℓ ˜ J q U ℓ ) j qq . Q ∗ M = Q M and M ∗ = M , we have Q M M = ( M Q M ) ∗ = O q × q . Consequently, Q M P M = Q M M M † = O q × q follows. In view of Notation 9.2, thus we obtain( Q M ⊕ O q × q ) W = ( Q M ⊕ O q × q ) " − yP M M ∗ ∗ = " − yQ M P M Q M MO q × q O q × q = O q × q . In particular, ( Q M ⊕ O q × q ) W ℓ = O q × q . Using ˜ J ∗ q = ˜ J q and Remark A.37, we get then U ∗ ℓ ˜ J q U ℓ = W ∗ ℓ ˜ J q W ℓ + 2 Re h x ( O q × q ⊕ Q M ) ˜ J q W ℓ i + xx ( O q × q ⊕ Q M ) ˜ J q ( O q × q ⊕ Q M )= W ∗ ℓ ˜ J q W ℓ + 2 Re h x ˜ J q ( Q M ⊕ O q × q ) W ℓ i + | x | ˜ J q ( Q M ⊕ O q × q )( O q × q ⊕ Q M ) = W ∗ ℓ ˜ J q W ℓ , implying V ∗ ℓ ˜ J q V ℓ = − j qq ( W ∗ ℓ ˜ J q W ℓ ) j qq . Remark A.37 yields j qq ( R ⊕ S ) j qq = R ⊕ S and j qq ( R ⊕ S ) ∗ ˜ J q ( R ⊕ S ) j qq = ( R ⊕ S ) ∗ ( j qq ˜ J q j qq )( R ⊕ S ) = − ( R ⊕ S ) ∗ ˜ J q ( R ⊕ S )for all R, S ∈ C q × q . The asserted identities can now be deduced from Proposition 9.4.We are now going to consider the composition of the transformations introduced in Defini-tions 9.1 and 9.5. Lemma 9.10.
Let M ∈ C p × q and let z ∈ C . Then h ¨ V M ( z ) ih ¨ W M ( z ) i = − ( β − z ) δ ( P M ⊕ I q ) = h ¨ W M ( z ) ih ¨ V M ( z ) i . Proof.
Let x := z − α and let y := β − z . We have then h ¨ V M ( z ) ih ¨ W M ( z ) i = yP M M − yxM † yI q − yP M M − yxM † − yP M ∗ − δQ M = − y P M − yxM M † yP M M − yM P M ∗ − δM Q M y xM † P M − y xM † − yxM † M − y P M ∗ − yδQ M and h ¨ W M ( z ) ih ¨ V M ( z ) i = − yP M M − yxM † − yP M ∗ − δQ M yP M M − yxM † yI q = − y P M − yxM M † − yP M M + yM − y xM † P M + y xP M ∗ M † + yxδQ M M † − yxM † M − y P M ∗ − yδQ M . Consequently, in view of (9.1), (9.2), and y + x = δ , the assertion follows.For a given non-negative Hermitian matrix M , the condition in Definition 9.5 is satisfiedfor pairs belonging to the subclass ¨ P [ M ] of PR q ( C \ [ α, β ]), introduced in Section 8. Hence,for suchlike pairs the corresponding inverse F α,β ( M )-transform exists and can be written as alinear fractional transformation, as considered in Appendix E:41 roposition 9.11. Let M ∈ C q × q < and let [ G ; G ] ∈ ¨ P [ M ] . In view of the functions g, h : C \ [ α, β ] → C defined by (9.9) , let F and F be given via (9.10) as matrix-valued functionsmeromorphic in C \ [ α, β ] . Then det F does not identically vanish in C \ [ α, β ] . Furthermore, det F ( z ) = 0 and F ( z ) = [ F ( z )][ F ( z )] − for all z ∈ C \ ([ α, β ] ∪ P ( G ) ∪ P ( G ) ∪ ¨ E ([ G ; G ])) ,where F denotes the inverse F α,β ( M ) -transform of [ G ; G ] .Proof. According to Notation 8.1, the pair [ G ; G ] belongs to PR q ( C \ [ α, β ]). Hence, G and G are C q × q -valued functions, which are meromorphic in C \ [ α, β ]. Furthermore, by virtue ofProposition 7.9, the set A := P ( G ) ∪ P ( G ) ∪ ¨ E ([ G ; G ]) is a discrete subset of C \ [ α, β ].Consequently, C \ ([ α, β ] ∪ A ) = ∅ . Consider an arbitrary z ∈ C \ ([ α, β ] ∪ A ). Then G and G are both holomorphic in z . Thus F and F are both holomorphic in z as well. Consideran arbitrary v ∈ N ( F ( z )). Regarding Remark A.16, we are going to show in a first step that k R [ G ( z )] v k E = 0 (9.11)holds true, where R := √ M † . Because of z = β , we have, according to (9.9) and (9.10), theequation ( z − α ) M † [ G ( z )] v = [ G ( z )] v. (9.12)In view of Remark A.14, hence v ∗ [ G ( z )] ∗ [ G ( z )] v = ( z − α ) v ∗ [ G ( z )] ∗ M † [ G ( z )] v = ( z − α ) k R [ G ( z )] v k . (9.13)In the case z ∈ C \ R , we see from Lemma 7.8 and Remark A.2 then that0 ≤ v ∗ (cid:18) z Im([ G ( z )] ∗ [ G ( z )]) (cid:19) v = 1Im z Im( v ∗ [ G ( z )] ∗ [ G ( z )] v )= 1Im z Im (cid:16) ( z − α ) k R [ G ( z )] v k (cid:17) = −k R [ G ( z )] v k ≤ , implying (9.11). If z ∈ ( −∞ , α ), then z = z < α and we obtain, by virtue of Lemma 7.8 and(9.13), thus 0 ≤ v ∗ [ G ( z )] ∗ [ G ( z )] v = ( z − α ) k R [ G ( z )] v k ≤ , implying again (9.11). In the case z ∈ ( β, ∞ ), we have z = z > β > α and, because ofLemma 7.8 and (9.13), similarly0 ≤ v ∗ ( − [ G ( z )] ∗ [ G ( z )]) v = − v ∗ [ G ( z )] ∗ [ G ( z )] v = ( α − z ) k R [ G ( z )] v k ≤ , i. e., (9.11). Thus, (9.11) is verified. Consequently, we can infer P M [ G ( z )] v = M M † [ G ( z )] v = M R [ G ( z )] v = O q × . In view of Lemma 8.3, we have furthermore P M G = G . Hence, [ G ( z )] v = O q × follows.Because of (9.12), this implies [ G ( z )] v = O q × . Observe that, due to Proposition 7.9, the q × q matrix pair [ G ( z ); G ( z )] is regular. According to Remark D.5, thus necessarily v = O q × holds true. Therefore, the linear subspace N ( F ( z )) is trivial, implying det F ( z ) = 0. Inparticular, det F does not identically vanish in C \ [ α, β ] and F ( z ) = [ F ( z )][ F ( z )] − .For any non-negative Hermitian matrix M , the inverse F α,β ( M )-transformation induces,according to Definition 7.11 and Remark 8.4, a well-defined transformation for equivalenceclasses from h ¨ P [ M ] i : 42 orollary 9.12. Let M ∈ C q × q < and let the pairs [ G ; G ] , [ ˜ G ; ˜ G ] ∈ ¨ P [ M ] be equivalent. Thenthe inverse F α,β ( M ) -transform F of [ G ; G ] coincides with the inverse F α,β ( M ) -transform ˜ F of [ ˜ G ; ˜ G ] .Proof. Using the functions g, h : C \ [ α, β ] → C given via (9.9), we define by (9.10) two C q × q -valued functions F and F meromorphic in C \ [ α, β ]. Because of Proposition 9.11,then det F does not vanish identically in C \ [ α, β ]. By virtue of Definition 9.5, we thushave F = F F − . Furthermore, due to Definition 7.11, there exists a C q × q -valued function R meromorphic in C \ [ α, β ] such that det R does not vanish identically in C \ [ α, β ] satisfying˜ G = G R and ˜ G = G R . Using again the functions g and h , we define according to (9.10)by ˜ F := hP M ˜ G + M ˜ G and ˜ F := − hgM † ˜ G + h ˜ G two C q × q -valued functions ˜ F and ˜ F meromorphic in C \ [ α, β ]. Then ˜ F = F R and ˜ F = F R . The application of Proposition 9.11to the pair [ ˜ G ; ˜ G ] yields furthermore that det ˜ F does not vanish identically in C \ [ α, β ]. Byvirtue of Definition 9.5, hence ˜ F = ˜ F ˜ F − . Consequently, ˜ F = ( F R )( F R ) − = F .Furthermore, after transition to equivalence classes as mentioned above, the F α,β ( M )-transformation turns out to be inverse to the inverse F α,β ( M )-transformation: Lemma 9.13.
Let M ∈ C q × q < and let [ G ; G ] ∈ ¨ P [ M ] with inverse F α,β ( M ) -transform F .Then [ G ; G ] is equivalent to the F α,β ( M ) -transformed pair of F .Proof. Using the functions g, h : C \ [ α, β ] → C given via (9.9), we define by (9.10) two C q × q -valued functions F and F meromorphic in C \ [ α, β ]. Then P M F = F . Denoteby [ H ; H ] the F α,β ( M )-transformed pair of F and by W the restriction of the holomorphic C q × q -valued function ¨ W M onto C \ [ α, β ]. In view of Definition 9.1 and Notation 9.2, then (cid:2) H H (cid:3) = − W (cid:2) FI q (cid:3) . Due to Proposition 9.11, the function det F does not vanish identically in C \ [ α, β ]. According to Definition 9.5, we thus have F = F F − . Denote by V the restrictionof the holomorphic C q × q -valued function ¨ V M onto C \ [ α, β ]. Regarding Notation 9.6, then (cid:2) F F (cid:3) = V (cid:2) G G (cid:3) . Taken all together, we get " H H = − W " F F − I q = − W " F F F − = − W V " G G F − = − W V " G F − G F − . From Lemma 8.3 we see P M G = G . Using Lemma 9.10, we thus can infer H = hδP M G F − = hδG F − and H = hδG F − . Observe that R := δhF − is a C q × q -valuedfunction, which is meromorphic in C \ [ α, β ] satisfying H = G R and H = G R . Furthermore,because of δ = 0, the function det R does not vanish identically in C \ [ α, β ]. According toDefinition 7.11, consequently [ G ; G ] ∼ [ H ; H ].Conversely, we have: Lemma 9.14.
Let M ∈ C q × q < and let F : C \ [ α, β ] → C q × q be a matrix-valued functionwith F α,β ( M ) -transformed pair [ G ; G ] such that P M F = F and [ G ; G ] ∈ ¨ P [ M ] hold true.Then the inverse F α,β ( M ) -transform of [ G ; G ] coincides with F .Proof. Since [ G ; G ] belongs to ¨ P [ M ], we see that G and G are C q × q -valued functions, whichare meromorphic in C \ [ α, β ]. Using the functions g, h : C \ [ α, β ] → C given via (9.9), we canthus define by (9.10) two C q × q -valued functions F and F , which then are meromorphic in C \ [ α, β ] as well. Denote by V the restriction of the holomorphic C q × q -valued function43 V M onto C \ [ α, β ]. From Notation 9.6 we see (cid:2) F F (cid:3) = V (cid:2) G G (cid:3) . Denote by W the restrictionof the holomorphic C q × q -valued function ¨ W M onto C \ [ α, β ]. Regarding Definition 9.1,Notation 9.2, and P M F = F , we have furthermore W (cid:2) FI q (cid:3) = − (cid:2) G G (cid:3) . Taken all together, weobtain (cid:2) F F (cid:3) = − V W (cid:2) FI q (cid:3) . In view of Lemma 9.10, thus F = hδP M F = hδF and F = hδI q follow. Taking into account [ G ; G ] ∈ ¨ P [ M ], we see from Proposition 9.11 that det F doesnot vanish identically in C \ [ α, β ]. Denote by H the inverse F α,β ( M )-transform of [ G ; G ].According to Definition 9.5, then H = F F − . Consequently, H = F .
10. The F α,β ( A, M ) -transformation and its inverse In this section, we continue the preceding considerations concerning the construction of thefunction-theoretic version of the Schur–Nevanlinna type algorithm. We prepare the algebraicformalism for the remaining steps after the first one.In what follows, we consider two complex q × q matrices A and M , which, in the context ofthe matricial Hausdorff moment problem, will later be the non-negative Hermitian matrices a and s for a given sequence ( s j ) κj =0 ∈ F < q,κ,α,β . In this reading, the matrices B := δM − A and N := A + αM (10.1)correspond to b and s , resp., and we have A = − αM + N and B = βM − N, (10.2)according to Notation 3.2. Consider an arbitrarily given z ∈ C and let x := z − α and y := β − z .Taking additionally into account (9.3) and (9.4), we then infer yA − xB = ( y + x ) N − ( αy + βx ) M = δ ( N − zM ) (10.3)and xA − yB = ( x + y ) N − ( αx + βy ) M = δ [ N − ( β + α − z ) M ] . (10.4) Definition 10.1.
Let G be a non-empty subset of C , let F : G → C p × q be a matrix-valuedfunction, and let A and M be two complex p × q matrices. Then G : G → C p × q defined by G ( z ) := AM † [( β − z ) F ( z ) − M ](( β − z )[( z − α ) F ( z ) + M ]) † A is called the F α,β ( A, M ) -transform of F .In connection with the F α,β ( A, M )-transformation, we consider the following complex( p + q ) × ( p + q ) matrix polynomial: Notation . Let A and M be two complex p × q matrices. Then let ¨ W A,M : C → C ( p + q ) × ( p + q ) be defined by ¨ W A,M ( z ) := − ( β − z ) AM † A − ( β − z )( z − α ) A † − ( β − z ) A † M − Q A . Under certain conditions, we can write the F α,β ( A, M )-transform as a linear fractional trans-formation with the generating matrix-valued function ¨ W A,M .44 emma 10.3.
Let
A, M ∈ C p × q be such that N ( M ) ⊆ N ( A ) . Let F : C \ [ α, β ] → C p × q bea matrix-valued function with F α,β ( A, M ) -transform G and let G , G : C \ [ α, β ] → C q × q bedefined by G ( w ) := − ( β − w ) AM † F ( w ) + A (10.5) and G ( w ) := − ( β − w )( w − α ) A † F ( w ) − ( β − w ) A † M − Q A . (10.6) Let z ∈ C \ [ α, β ] be such that R ( F ( z )) ⊆ R ( M ) and N ( M ) ⊆ N ( F ( z )) as well as R (( z − α ) F ( z ) + M ) = R ( A ) and N (( z − α ) F ( z ) + M ) = N ( A ) are fulfilled. Then det G ( z ) = 0 and G ( z ) = [ G ( z )][ G ( z )] − .Proof. Let X := ( z − α ) F ( z ) + M and let Y := ( β − z ) F ( z ) − M . From Remark A.21 we get AM † M = A . Remark A.18 shows that I q − A † A = Q A . Setting y := β − z and Z := yX , weobtain then − AM † Y = − yAM † F ( z ) + AM † M = − yAM † F ( z ) + A = G ( z ) (10.7)and − A † Z − ( I q − A † A ) = − yA † X − Q A = G ( z ) . (10.8)Using Remark A.22, we get XM † Y = Y M † X . In view of Remark A.18, furthermore X † X = A † A holds true. Thus, we can infer AM † Y = AA † AM † Y = AX † XM † Y = AX † Y M † X and, therefore, AM † Y A † A = AX † Y M † XA † A = AX † Y M † XX † X = AX † Y M † X = AM † Y. Regarding y = 0, we have R ( Z ) = R ( A ) and N ( Z ) = N ( A ). By virtue of Remark A.18, hence ZZ † = AA † and Z † Z = A † A . Consequently, h − Z † A − ( I q − A † A ) ih − A † Z − ( I q − A † A ) i = Z † AA † Z + Z † A ( I q − A † A ) + ( I q − A † A ) A † Z + ( I q − A † A ) = Z † ZZ † Z + I q − A † A = Z † Z + I q − Z † Z = I q . Hence, det[ − A † Z − ( I q − A † A )] = 0 and [ − A † Z − ( I q − A † A )] − = − Z † A − ( I q − A † A ). Thus, − AM † Y h − A † Z − ( I q − A † A ) i − = AM † Y h Z † A + ( I q − A † A ) i = AM † Y Z † A + AM † Y ( I q − A † A ) = AM † Y Z † A = G ( z ) . In view of (10.7) and (10.8), the proof is complete.From Lemmata 10.12 and 10.13 we will see that the following transformation for matrix-valued functions is in a generic situation indeed the inversion of the F α,β ( A, M )-transformation.Against this background we introduce the following notation:45 efinition 10.4.
Let G be a non-empty subset of C , let G : G → C p × q be a matrix-valuedfunction, and let A and M be two complex p × q matrices. Let B := δM − A and let F : G → C p × q be defined by F ( z ) := − h ( β − z ) M A † G ( z ) + A + M Q A M † B i × (cid:16) ( β − z ) h ( z − α ) A † G ( z ) − M † A i + ( z − α ) Q A M † B (cid:17) † . Then we call the matrix-valued function F the inverse F α,β ( A, M ) -transform of G . Lemma 10.5.
Let A ∈ C q × q H and let M ∈ C q × q < with R ( A ) ⊆ R ( M ) . Let G ∈ R q ( C \ [ α, β ]) be such that R ( G ( z )) ⊆ R ( A ) holds true for some z ∈ C \ [ α, β ] . Let B := δM − A andlet E , E : C \ [ α, β ] → C q × q be defined by E ( w ) := ( β − w ) M A † G ( w ) + A + M Q A M † B (10.9) and E ( w ) := − ( β − w )( w − α ) A † G ( w ) + ( β − w ) M † A − ( w − α ) Q A M † B. (10.10) For all z ∈ C \ [ α, β ] , then R ( E ( z )) ⊆ R ( M ) , N ( M ) ⊆ N ( E ( z )) , R ( E ( z )) = R ( M ) ,and N ( E ( z )) = N ( M ) . Furthermore, the inverse F α,β ( A, M ) -transform of G is holomorphicin C \ [ α, β ] .Proof. Consider an arbitrary z ∈ C \ [ α, β ]. Regarding Remark A.6, we have R ( E ( z )) ⊆R ( M A † G ( z )) + R ( A ) + R ( M Q A M † B ) ⊆ R ( M ). From Proposition 4.15 we infer R ( G ( z )) = R ( G ( z )) ⊆ R ( A ) ⊆ R ( M ). Consequently, [ R ( M )] ⊥ ⊆ [ R ( A )] ⊥ ⊆ [ R ( G ( z ))] ⊥ . In view ofRemarks A.10 and 4.8, then N ( M ) ⊆ N ( A ) ⊆ N ([ G ( z )] ∗ ) = N ( G ( z )) (10.11)follows. Regarding N ( M ) ⊆ N ( A ) and Remark A.6, we can conclude N ( M ) ⊆ N ( B )and furthermore N ( M ) ⊆ N ( M A † G ( z )) ∩ N ( A ) ∩ N ( M Q A M † B ) ⊆ N ( E ( z )). By virtueof Remark A.14, we have R ( A † ) = R ( A ) ⊆ R ( M ) = R ( M † ). In view of Remark A.14,then M † M A † = A † holds true, according to Remark A.20. Since Remark A.18 shows that Q A = I q − A † A , we infer in particular Q A M † = M † ( I q − M A † AM † ). Therefore, we ob-tain R ( Q A M † ) ⊆ R ( M † ) and, regarding additionally Remark A.6, furthermore R ( E ( z )) ⊆R ( A † G ( z )) + R ( M † A ) + R ( Q A M † B ) ⊆ R ( M † ) = R ( M ). In view of Remark A.16, let R := √ M † A . Consider an arbitrary v ∈ N ( E ( z )). We are now going to check that k Rv k E = 0 (10.12)holds true. Obviously,( β − z ) M † Av = ( β − z )( z − α ) A † [ G ( z )] v + ( z − α ) Q A M † Bv. (10.13)In view of Proposition 4.15, we have R ( G ( z )) = R ( G ( z )) ⊆ R ( A ). According to Remark A.20,thus AA † G ( z ) = G ( z ). Regarding z = β , we can multiply equation (10.13) from the left by( β − z ) − A to obtain then AM † Av = ( z − α )[ G ( z )] v . Left multiplication of this identity by( z − α ) v ∗ yields | z − α | v ∗ [ G ( z )] v = ( z − α ) v ∗ AM † Av = ( z − α ) k Rv k . (10.14)46n the case z ∈ C \ R , we can infer from Proposition 4.14 and Remark A.2 that0 ≤ | z − α | v ∗ (cid:20) z Im G ( z ) (cid:21) v = 1Im z Im (cid:16) | z − α | v ∗ [ G ( z )] v (cid:17) = 1Im z Im h ( z − α ) k Rv k i = −k Rv k ≤ , implying (10.12). If z ∈ ( −∞ , α ), then z = z < α and we obtain, by virtue of Notation 4.6and (10.14), thus 0 ≤ | z − α | v ∗ [ G ( z )] v = ( z − α ) k Rv k ≤
0, implying again (10.12). In thecase z ∈ ( β, ∞ ), we have z = z > β > α and, because of Notation 4.6 and (10.14), similarly0 ≤ | z − α | v ∗ [ − G ( z )] v = −| z − α | v ∗ [ G ( z )] v = ( α − z ) k Rv k ≤ , i. e. (10.12). Hence, (10.12) is verified. Consequently, using Remark A.20, we can infer Av = M M † Av = M √ M † Rv = O q × . Regarding (10.11), we thus obtain [ G ( z )] v = O q × . Becauseof (10.11) and Remark A.21, we have AM † M = A . In view of Q A = I q − A † A , in particular Q A M † M = M † M − A † A holds true. Taking into account Av = O q × and [ G ( z )] v = O q × , wesee from (10.13) then O q × = ( z − α ) Q A M † Bv = ( z − α ) Q A M † ( δM ) v = ( z − α ) δM † M v.
Left multiplication of the latter by M yields ( z − α ) δM v = O q × . Since z = α and δ > M v = O q × follows. Hence, N ( E ( z )) ⊆ N ( M ). Because of (10.11) and N ( M ) ⊆ N ( B ), we obtain, by virtue of Remark A.6, on the other hand N ( M ) ⊆ N ( A † G ( z )) ∩N ( M † A ) ∩ N ( Q A M † B ) ⊆ N ( E ( z )). Consequently, N ( E ( z )) = N ( M ) is verified. UsingRemark A.3, we can, in view of R ( E ( z )) ⊆ R ( M ), then easily conclude R ( E ( z )) = R ( M ).Observe that the matrix-valued function G is holomorphic. Thus, E and E are holomorphicin C \ [ α, β ] as well. Let D : C \ [ α, β ] → C q × q be defined by D ( w ) := [ E ( w )] † . As alreadyshown, the linear subspaces R ( E ( w )) and N ( E ( w )) do not depend on the point w ∈ C \ [ α, β ].Due to Proposition F.4, thus the matrix-valued function D is holomorphic. Denote by F theinverse F α,β ( A, M )-transform of G . In view of Remarks A.15 and A.13, we have E D = F .Using Remark F.2, we can conclude then that the matrix-valued function F is holomorphic in C \ [ α, β ].The following complex ( p + q ) × ( p + q ) matrix polynomial is intimately connected to theinverse F α,β ( A, M )-transform:
Notation . Let A and M be two complex p × q matrices and let B := δM − A . Then let¨ V A,M : C → C ( p + q ) × ( p + q ) be defined by¨ V A,M ( z ) := ( β − z ) M A † A + M Q A M † B − ( β − z )( z − α ) A † ( β − z )( δQ M + M † A ) − ( z − α ) Q A M † B . Remark . Let
A, M ∈ C q × q and let z ∈ C . Let B := δM − A . Then:(a) If A = O q × q , then B = δM and ¨ V A,M ( z ) = δ h O q × q MO q × q ( β − z ) I q − δM † M i .(b) If B = O q × q , then A = δM and ¨ V A,M ( z ) = h δ − ( β − z ) MM † δM − δ − ( β − z )( z − α ) M † δ ( β − z ) I q i .(c) If all the matrices M, A, B are invertible, then ¨ V A,M ( z ) = h ( β − z ) MA − A − ( β − z )( z − α ) A − ( β − z ) M − A i .47nder certain conditions, we can write the inverse F α,β ( A, M )-transform as a linear frac-tional transformation with the generating matrix-valued function ¨ V A,M . Lemma 10.8.
Let A ∈ C q × q H and let M ∈ C q × q < with R ( A ) ⊆ R ( M ) . Let G ∈ R q ( C \ [ α, β ]) with inverse F α,β ( A, M ) -transform F be such that R ( G ( z )) ⊆ R ( A ) holds true for some z ∈ C \ [ α, β ] . Let B := δM − A and let F , F : C \ [ α, β ] → C q × q be defined by F ( w ) := ( β − w ) M A † G ( w ) + A + M Q A M † B (10.15) and F ( w ) := − ( β − w )( w − α ) A † G ( w ) + ( β − w )( δQ M + M † A ) − ( w − α ) Q A M † B. (10.16) For all z ∈ C \ [ α, β ] , then det F ( z ) = 0 and F ( z ) = [ F ( z )][ F ( z )] − .Proof. Let E , E : C \ [ α, β ] → C q × q be defined by (10.9) and (10.10). Consider an arbitrary z ∈ C \ [ α, β ]. We have F ( z ) = E ( z ) and F ( z ) = E ( z ) + ( β − z ) δQ M . From Lemma 10.5 we get R ( E ( z )) = R ( M ) and N ( E ( z )) = N ( M ). Remark A.10 yields then R ( E ( z )) = R ( M ∗ ) = R ([ E ( z )] ∗ ). In view of z = β and δ >
0, we thus can apply Lemma A.19 with η := ( β − z ) δ tosee that the matrix E ( z ) + ηQ M = F ( z ) is invertible and that [ E ( z )] † = [ F ( z )] − − η − Q M holds true. By virtue of Lemma 10.5, we have N ( M ) ⊆ N ( E ( z )). Consequently, we obtain[ F ( z )][ F ( z )] − = [ E ( z )] (cid:16) [ E ( z )] † + η − Q M (cid:17) = [ E ( z )][ E ( z )] † = F ( z ) . Lemma 10.9.
Let A ∈ C q × q H and let M ∈ C q × q < with R ( A ) ⊆ R ( M ) . Let G ∈ R q ( C \ [ α, β ]) with inverse F α,β ( A, M ) -transform F . Suppose that R ( G ( z )) ⊆ R ( A ) holds true for some z ∈ C \ [ α, β ] . For all z ∈ C \ [ α, β ] , then R ( F ( z )) ⊆ R ( M ) and N ( M ) ⊆ N ( F ( z )) andfurthermore R (( z − α ) F ( z ) + M ) = R ( A ) and N (( z − α ) F ( z ) + M ) = N ( A ) .Proof. Consider an arbitrary z ∈ C \ [ α, β ]. Let E , E : C \ [ α, β ] → C q × q be defined by (10.9)and (10.10). According to Definition 10.4, then F ( z ) = [ E ( z )][ E ( z )] † . Lemma 10.5 yieldsfurthermore R ( E ( z )) ⊆ R ( M ) and R ( E ( z )) = R ( M ). By virtue of Remarks A.14 and A.10,we can infer from the last identity N ([ E ( z )] † ) = N ( M ∗ ) = N ( M ). Consequently, we obtain R ( F ( z )) ⊆ R ( M ) and N ( M ) ⊆ N ( F ( z )). Taking additionally into account Remark A.14, wecan conclude M [ E ( z )][ E ( z )] † = M . Remark A.20 yields M M † A = A . Let x := z − α , let y := β − z , and let X := xF ( z ) + M . Taken all together, we get X = [ xE ( z ) + M E ( z )][ E ( z )] † = ( xA + yM M † A )[ E ( z )] † = δA [ E ( z )] † . Analogous to the corresponding considerations in the proof of Lemma 10.8, we can show thatthe matrix R := E ( z ) + Q M is invertible and that [ E ( z )] † = R − − Q M holds true. As inthe proof of Lemma 10.5, we can obtain (10.11). Thus, X = δA ( R − − Q M ) = δAR − follows.Regarding δ >
0, we see from Remark A.8 hence R ( X ) = R ( A ) and N ( X ) = R N ( A ). Let B := δM − A . In view of (10.10) and (10.11), each v ∈ N ( A ) satisfies Rv = − xQ A M † Bv + Q M v and thus ARv = O q × . Consequently, N ( X ) ⊆ N ( A ) is verified. Taking additionally intoaccount R ( X ) = R ( A ), we infer by virtue of Remark A.3 then easily N ( X ) = N ( A ).Now we are going to study the composition of the two transformations introduced in Def-initions 10.1 and 10.4. Doing this, we will take into account that, in view of Lemmata 10.3and 10.8, these transformations can be written under certain conditions as linear fractionaltransformations of matrices with generating matrix-valued functions ¨ W A,M and ¨ V A,M , resp.48 emma 10.10.
Let
A, M ∈ C p × q with R ( A ) ⊆ R ( M ) and N ( M ) ⊆ N ( A ) , let B := δM − A ,and let N := A + αM . Let z ∈ C , let x := z − α , and let y := β − z . Then h ¨ W A,M ( z ) ih ¨ V A,M ( z ) i = − δ (cid:16) [ yP A ] ⊕ h yP A ∗ − βQ M + Q A ( zI q − M † N ) i(cid:17) and yP A ∗ − βQ M + Q A ( zI q − M † N ) = ( M † B + P A ∗ M † A ) − yQ A − xP A ∗ . (10.17) Proof.
For the block representation [ ¨ W A,M ( z )][ ¨ V A,M ( z )] = (cid:2) X X X X (cid:3) with p × p block X , wehave X = − yAM † ( yM A † ) + A ( − yxA † ) ,X = − yAM † ( A + M Q A M † B ) + A h y ( δQ M + M † A ) − xQ A M † B i ,X = − yxA † ( yM A † ) − ( yA † M − Q A )( − yxA † ) , and X = − yxA † ( A + M Q A M † B ) − ( yA † M − Q A ) h y ( δQ M + M † A ) − xQ A M † B i . The application of Remarks A.20 and A.21 yields
M M † A = A and AM † M = A . Regarding(9.1), (9.2), and y + x = δ , we obtain then X = − y AM † M A † − yxAA † = − y ( yAA † + xAA † ) = − yδP A ,X = − yAM † A − yAM † M Q A M † B + yAM † A = − yAQ A M † B = O p × q ,X = − y xA † M A † + y xA † M A † + yxQ A A † = yx ( I q − A † A ) A † = O q × p , and X = − yxA † A − yxA † M Q A M † B − y A † M M † A + yxA † M Q A M † B + yδQ A Q M + yQ A M † A − xQ A M † B = − yxA † A − y A † A + yδQ M + yQ A M † A − xQ A M † B = − yδP A ∗ + ( β − z ) δQ M + yQ A M † A − xQ A M † B. (10.18)In view of (10.3), we have yM † A − xM † B = M † ( yA − xB ) = δM † ( N − zM ) . By virtue of Q M = I q − M † M , we thus get yM † A − xM † B − zδQ M = δ ( M † N − zM † M − zI q + zM † M ) = δ ( M † N − zI q ) . (10.19)From (10.18) we can infer then X + yδP A ∗ − βδQ M = − zδQ M + yQ A M † A − xQ A M † B = yQ A M † A − xQ A M † B − zδQ A Q M = δQ A ( M † N − zI q ) , i. e., X = − δ [ yP A ∗ − βQ M + Q A ( zI q − M † N )]. Because of (9.2), we have M † B = M † ( δM − A ) = δM † M − M † A = δP M ∗ − M † A,P A ∗ − Q M = A † A − ( I q − M † M ) = M † M − ( I q − A † A ) = P M ∗ − Q A , AM † M = A , furthermore Q A P M ∗ = ( I q − A † A ) M † M = M † M − A † A = P M ∗ − P A ∗ . From (10.18) we thus can conclude X = − y h δ ( P A ∗ − Q M ) − Q A M † A i − xQ A ( δP M ∗ − M † A )= − y h δ ( P M ∗ − Q A ) − Q A M † A i − x h δ ( P M ∗ − P A ∗ ) − Q A M † A i = − ( y + x )( δP M ∗ − Q A M † A ) + yδQ A + xδP A ∗ = − δ ( δP M ∗ − M † A + A † AM † A ) + δ ( yQ A + xP A ∗ )= − δ h ( M † B + P A ∗ M † A ) − yQ A − xP A ∗ i . Comparing the two representations of X , we can infer then (10.17).The next result is concerned with the matrix polynomial from (10.17): Lemma 10.11.
Let
A, M ∈ C p × q with N ( M ) ⊆ N ( A ) and let z ∈ C \ [ α, β ] . Let N := A + αM ,let y := β − z , and let H := yP A ∗ − βQ M + Q A ( zI q − M † N ) . Then det H = 0 and AH − = y − A .Proof. Consider an arbitrary v ∈ N ( H ). We have then βQ M v − yA † Av = ( βQ M − yP A ∗ ) v = Q A ( zI q − M † N ) v = ( I q − A † A )( zI q − M † N ) v. (10.20)Regarding z = β , left multiplication of the latter identity by − y − A yields Av = O p × . Conse-quently, N v = αM v . Taking into account Remark A.21, thus AM † N v = αAM † M v = αAv = O p × . From (10.20) we then infer βQ M v = ( I q − A † A )( zI q − M † N ) v = ( zI q − M † N ) v = zv − αM † M v. (10.21)Left multiplying this by M , we get O p × = ( z − α ) M v . Since z = α , then necessarily M v = O p × . Substituting this into (10.21) and regarding Q M = I q − M † M , we obtain βv = zv .Because of z = β , hence v = O p × follows. Consequently, the linear subspace N ( H ) is trivial,implying det H = 0. By virtue of AH = yAP A ∗ = yAA † A = yA and z = β , we thus have AH − = y − A .In generic situations, the F α,β ( A, M )-transformation turns out to be inverse to the inverse F α,β ( A, M )-transformation:
Lemma 10.12.
Let A ∈ C q × q H and let M ∈ C q × q < with R ( A ) ⊆ R ( M ) . Let G ∈ R q ( C \ [ α, β ]) with inverse F α,β ( A, M ) -transform F . Suppose that R ( G ( z )) ⊆ R ( A ) holds true for some z ∈ C \ [ α, β ] . Then G is exactly the F α,β ( A, M ) -transform of F .Proof. Consider an arbitrary z ∈ C \ [ α, β ]. Let F , F : C \ [ α, β ] → C q × q be defined by (10.15)and (10.16). Then (cid:2) F ( z ) F ( z ) (cid:3) = [ ¨ V A,M ( z )] (cid:2) G ( z ) I q (cid:3) . Due to Lemma 10.8, furthermore det F ( z ) = 0and F ( z ) = [ F ( z )][ F ( z )] − . Denote by H the F α,β ( A, M )-transform of F and let H , H : C \ [ α, β ] → C q × q be defined by H ( w ) := − ( β − w ) AM † F ( w ) + A H ( w ) := − ( β − w )( w − α ) A † F ( w ) − ( β − w ) A † M − Q A . Using Remark A.10, we infer N ( M ) ⊆ N ( A ). In view of Lemma 10.9, we thus can applyLemma 10.3 to F to obtain det H ( z ) = 0 and H ( z ) = [ H ( z )][ H ( z )] − . Since by construction (cid:2) H ( z ) H ( z ) (cid:3) = [ ¨ W A,M ( z )] (cid:2) F ( z ) I q (cid:3) holds true, we have " H ( z ) H ( z ) = h ¨ W A,M ( z ) i " F ( z ) F ( z ) [ F ( z )] − = h ¨ W A,M ( z ) i [ ¨ V A,M ( z )] " G ( z ) I q [ F ( z )] − . Let x := z − α , let y := β − z , and let N := A + αM . Taking into account Lemma 10.10, then H ( z ) = − δyP A [ G ( z )][ F ( z )] − , H ( z ) = − δ h yP A ∗ − βQ M + Q A ( zI q − M † N ) i [ F ( z )] − follow by comparing both sides of the latter identity. According to Proposition 4.15, we get R ( G ( z )) = R ( G ( z )) ⊆ R ( A ). Thus, P A G ( z ) = G ( z ). Using again Proposition 4.15, wecan conclude R ( G ( z )) = R ( G ( z )) ⊆ R ( A ), implying [ R ( A )] ⊥ ⊆ [ R ( G ( z ))] ⊥ . In view ofRemarks A.10 and 4.8, we then obtain N ( A ) ⊆ N ([ G ( z )] ∗ ) = N ( G ( z )). Due to Remark A.21,therefore [ G ( z )] A † A = G ( z ) holds true. Regarding det H ( z ) = 0 and Lemma 10.11, we havefurthermore − δA [ F ( z )] − [ H ( z )] − = A (cid:16) − δ − [ H ( z )][ F ( z )] (cid:17) − = y − A. Consequently, H ( z ) = [ H ( z )][ H ( z )] − = − δy [ G ( z )] A † A [ F ( z )] − [ H ( z )] − = [ G ( z )] A † A = G ( z ) . Conversely, we have:
Lemma 10.13.
Let A ∈ C q × q H and let M ∈ C q × q < with R ( A ) ⊆ R ( M ) . Let F : C \ [ α, β ] → C q × q with F α,β ( A, M ) -transform G and denote by H the inverse F α,β ( A, M ) -transform of G . Let z ∈ C \ [ α, β ] be such that R ( F ( z )) ⊆ R ( M ) and N ( M ) ⊆ N ( F ( z )) as well as R (( z − α ) F ( z ) + M ) = R ( A ) and N (( z − α ) F ( z ) + M ) = N ( A ) are fulfilled. Suppose that G belongs to R q ( C \ [ α, β ]) and that R ( G ( z )) ⊆ R ( A ) holds true for some z ∈ C \ [ α, β ] . Then H ( z ) = F ( z ) .Proof. Because of Remark A.10, we have N ( M ) ⊆ N ( A ), implying AM † M = A , by virtue ofRemark A.21. Let G , G : C \ [ α, β ] → C q × q be defined by (10.5) and (10.6). The applicationof Lemma 10.3 yields then det G ( z ) = 0 and G ( z ) = [ G ( z )][ G ( z )] − . Let x := z − α and let y := β − z . Setting X := xF ( z ) + M and Y := yF ( z ) − M , we get G ( z ) = − yAM † F ( z ) + A = − yAM † F ( z ) + AM † M = − AM † Y and G ( z ) = − yxA † F ( z ) − yA † M − Q A = − yA † X − Q A . Taking into account the assumptions, we have XQ A = O q × q and, in view of Remarks A.20and A.21, furthermore M M † A = A , AA † X = X , and XA † A = X . Let B := δM − A and let51 , E : C \ [ α, β ] → C q × q be defined by (10.9) and (10.10). Since AQ A = O q × q holds obviouslytrue, we can infer then F ( z ) E ( z ) − E ( z )= F ( z ) h − yxA † G ( z ) + yM † A − xQ A M † B i − h yM A † G ( z ) + A + M Q A M † B i = − yx [ F ( z )] A † G ( z ) + y [ F ( z )] M † A − x [ F ( z )] Q A M † B − yM A † G ( z ) − M M † A − M Q A M † B = − yXA † G ( z ) + Y M † A − XQ A M † B = − yXA † [ G ( z )][ G ( z )] − + Y M † A [ G ( z )][ G ( z )] − = h − yXA † ( − AM † Y ) + Y M † A ( − yA † X − Q A ) i [ G ( z )] − = y ( XM † Y − Y M † X )[ G ( z )] − . Regarding Remark A.22, we see XM † Y = Y M † X . Consequently, F ( z ) E ( z ) = E ( z ) follows.Observe that H ( z ) = [ E ( z )][ E ( z )] † . Due to Lemma 10.5, we have R ( E ( z )) = R ( M ). UsingRemarks A.14 and A.10, we thus can conclude N ([ E ( z )] † ) = N ( M ∗ ) = N ( M ). In view of N ( M ) ⊆ N ( F ( z )), hence N ([ E ( z )] † ) ⊆ N ( F ( z )). Because of Remarks A.21 and A.14, then F ( z )[ E ( z )][ E ( z )] † = F ( z ) holds true. Consequently, we obtain H ( z ) = [ E ( z )][ E ( z )] † = F ( z )[ E ( z )][ E ( z )] † = F ( z ) . In the particular completely degenerate situation B = O p × q we have A = δM , according to(10.1). Because of (9.2), then the matrix polynomials ¨ V A,M and ¨ W A,M from Notations 10.6and 10.2 essentially coincide with ¨ V M and ¨ W M introduced in Notations 9.6 and 9.2, resp.: Remark . If M ∈ C p × q , then the equations ¨ V δM,M = ¨ V M [( δ − I p ) ⊕ ( δI q )] and ¨ W δM,M =[( δ − I p ) ⊕ ( δI q )] ¨ W M hold true.In addition to the matrices A and M and the matrices B and N built from them via (10.1),we now consider the matrix D := AM † B , which, in view of Remark 3.23, corresponds to d .Because of (10.2), we have D = ( − αM + N ) M † ( βM − N ) = − αβM + αP M N + βN P M ∗ − N M † N in analogy to the second equation in (3.7). Taking into account (10.1) and (3.9), we getfurthermore D = A (cid:20) δ ( A + B ) (cid:21) † B = δ h A ( A + B ) † B i = δ ( A ⊤−⊥ B ) . (10.22) Notation . Let
A, M ∈ C p × q and let B := δM − A and D := AM † B . Then let ¨ U A,M : C → C ( p + q ) × ( p + q ) be defined by¨ U A,M ( z ) := M [( β − z ) P A ∗ M † B + ( z − α ) Q A M † A ] D † B − ( β − z )( z − α ) M † AD † ( β − z )( δQ M + M † A ) . Remark . Let
A, M ∈ C q × q and let z ∈ C . Let B := δM − A and D := AM † B . Then:(a) If D = O q × q , then ¨ U A,M ( z ) = h O q × q BO q × q ( β − z )( δQ M + M † A ) i .(b) If A = O q × q , then B = δM , D = O q × q , and ¨ U A,M ( z ) = δ h O q × q MO q × q ( β − z ) Q M i .(c) If B = O q × q , then A = δM , D = O q × q and ¨ U A,M ( z ) = δ h O q × q O q × q O q × q ( β − z ) I q i .52d) If the matrices M, A, B, D are invertible and AM − B = BM − A , then ¨ U A,M ( z ) = h ( β − z ) MA − B − ( β − z )( z − α ) B − ( β − z ) M − A i . Lemma 10.17.
Let
A, M ∈ C q × q H with R ( A ) ⊆ R ( M ) and let z ∈ C . Let B := δM − A , D := AM † B , and N := A + αM and let x := z − α and y := β − z . Then h ¨ V A,M ( z ) ih ¨ V D ( z ) i = − yδ [( z − α − β ) M + N − M Q A M † N ] D † − Mx ( yI q + Q A M † N ) D † zI q − M † N − βQ M = h ¨ V M ( z ) ih ¨ U A,M ( z ) i . (10.23) Proof.
Because of Remark A.6, we have R ( B ) ⊆ R ( M ) and R ( N ) ⊆ R ( M ). Consequently,Remark A.20 shows M M † A = A , M M † B = B , and M M † N = N . In view of Q A = I q − A † A ,then M Q A M † N = M M † N − M A † AM † N = N − M A † AM † N (10.24)and, by virtue of Q A M † B = M † B − A † AM † B = M † B − A † D, (10.25)furthermore M Q A M † B = M M † B − M A † D = B − M A † D (10.26)follow. Taking into account A + B = δM and δ >
0, we can infer R ( A ) ⊆ R ( A + B ). Since A and M are Hermitian, Remark A.24 shows that B is Hermitian as well. Using [44, Thm. 2.2(b)], wecan thus conclude ( A ⊤−⊥ B ) ∗ = A ⊤−⊥ B . In view of (10.22), then D ∗ = D follows. Furthermore, wehave R ( D ) ⊆ R ( A ) ⊆ R ( M ). Using Remark A.14, we obtain then R ( D † ) ⊆ R ( A † ) ⊆ R ( M † ).From Remarks A.20 and A.14, we can thus conclude M † M D † = D † , M † M A † = A † , and A † AD † = D † . (10.27)In particular, A † AM † M D † = A † AD † = D † = M † M D † follows. In view of (10.2), we infer then A † DD † = A † AM † BD † = A † AM † ( βM − N ) D † = ( βM † M − A † AM † N ) D † . (10.28)Taking into account P D = DD † and Notations 10.6 and 9.6, the computation of the q × q ma-trices in the block representation [ ¨ V A,M ( z )][ ¨ V D ( z )] = (cid:2) X X X X (cid:3) yields X = y M A † P D − yx ( A + M Q A M † B ) D † = y ( yM A † D − xA − xM Q A M † B ) D † ,X = yM A † D + y ( A + M Q A M † B ) = y ( M A † D + A + M Q A M † B ) ,X = − y xA † P D − yx h y ( δQ M + M † A ) − xQ A M † B i D † = − yx ( yA † D + yδQ M + yM † A − xQ A M † B ) D † , and X = − yxA † D + y h y ( δQ M + M † A ) − xQ A M † B i = − y ( xA † D − yδQ M − yM † A + xQ A M † B ) .
53y virtue of (10.26), (10.28), and (10.24), we obtain X = y ( yM A † D − xA − xB + xM A † D ) D † = y ( δM A † D − xδM ) D † = − yδ ( xM − M A † D ) D † = − yδ h ( z − α ) M − M ( βM † M − A † AM † N ) i D † = − yδ h ( z − α − β ) M + M A † AM † N i D † = − yδ h ( z − α − β ) M + N − M Q A M † N i D † . Because of (10.26), we have X = y ( M A † D + A + B − M A † D ) = yδM = − yδ ( − M ) . Using (10.3), we get (10.19) by the same reasoning as in the proof of Lemma 10.10. Thecombination of (10.25), (10.19), (10.28), and (9.2) yields X = − yx h yA † D + ( β − z ) δQ M + yM † A − xM † B + xA † D i D † = − yx ( δA † D + βδQ M + yM † A − xM † B − zδQ M ) D † = − yx h δA † D + βδQ M + δ ( M † N − zI q ) i D † = − yxδ h βM † M − A † AM † N + β ( I q − M † M ) + M † N − zI q i D † = − yxδ ( yI q + Q A M † N ) D † . Taking into account (10.25) and (10.19), we get furthermore X = − y h xA † D − ( β − z ) δQ M − yM † A + xM † B − xA † D i = − y ( zδQ M − yM † A + xM † B − βδQ M ) = − yδ ( zI q − M † N − βQ M ) . Hence, the first equation in (10.23) is verified.Because of (10.4) and (10.27), we have
M A † AM † ( yB − xA ) D † = δM A † AM † [( β + α − z ) M − N ] D † = δ h ( β + α − z ) M − M A † AM † N i D † . (10.29)Taking into account P M = M M † and Notations 9.6 and 10.15, the computation of the q × q ma-trices in the block representation [ ¨ V M ( z )][ ¨ U A,M ( z )] = (cid:2) Y Y Y Y (cid:3) yields Y = yP M M ( yP A ∗ M † B + xQ A M † A ) D † − yxM M † AD † = y ( yM P A ∗ M † B + xM Q A M † A − xM M † A ) D † ,Y = yP M B + yM ( δQ M + M † A ) = yP M ( B + A ) = yP M ( δM ) = yδM = − yδ ( − M ) ,Y = − yxM † M ( yP A ∗ M † B + xQ A M † A ) D † − y xM † AD † = − yxM † ( yM P A ∗ M † B + xM Q A M † A + yA ) D † , and Y = − yxM † B + y ( δQ M + M † A ) = − y h xM † B − ( β − z ) δQ M − yM † A i .
54n view of P A ∗ = A † A and Q A = I q − A † A , we can infer from (10.29) and (10.24) then Y = y ( yM A † AM † B + xM M † A − xM A † AM † A − xM M † A ) D † = yM A † AM † ( yB − xA ) D † = − yδ h ( z − α − β ) M + M A † AM † N i D † = − yδ h ( z − α − β ) M + N − M Q A M † N i D † . Because of
M M † A = A and the identities (10.29) and (10.27), we have furthermore Y = − yxM † ( yM A † AM † B + xM M † A − xM A † AM † A + yA ) D † = − yxM † h M A † AM † ( yB − xA ) D † + xAD † + yAD † i = − yxM † (cid:16) δ h ( β + α − z ) M − M A † AM † N i D † + δAD † (cid:17) = − yxδ h ( y + α ) M † M D † − M † M A † AM † N D † + M † AD † i = − yxδ ( yD † + αM † M D † − A † AM † N D † + M † AD † )= − yxδ h yI q + M † ( αM + A ) − A † AM † N i D † = − yxδ ( yI q + Q A M † N ) D † . From (10.19) moreover Y = − yδ ( zI q − M † N − βQ M ) follows. By virtue of Y = − yδ ( − M ),thus the second equation in (10.23) is verified.
11. On the elementary steps of the forward algorithm
This section is aimed to work out the elementary step of the forward algorithm by applyingthe transformations studied in the previous section.
Lemma 11.1.
Let ( s j ) j =0 ∈ F < q, ,α,β and let F ∈ R q [[ α, β ]; ( s j ) j =0 ] . Then the F α,β ( s ) -transformed pair of F belongs to ¨ P [ s ] .Proof. Denote by [ G ; G ] the F α,β ( s )-transformed pair of F . Obviously, D := ∅ is a discretesubset of C \ [ α, β ]. Observe that F is holomorphic. In view of Definition 9.1, then G and G are holomorphic as well. In particular, G and G are C q × q -valued functions, which aremeromorphic in C \ [ α, β ] with P ( G ) ∪P ( G ) ⊆ D . Consequently, condition (I) in Notation 7.5is fulfilled with the set D for the pair [ P ; Q ] = [ G ; G ]. Consider an arbitrary z ∈ C \ [ α, β ]. Byassumption, the R [ α, β ]-measure ¨ σ F of F belongs to M < q, [[ α, β ]; ( s j ) j =0 , =], i. e., ¨ σ F ([ α, β ]) = s . Taking additionally into account Proposition 4.15, hence R ( F ( z )) = R ( s ) follows. Thus, P R ( s ) F ( z ) = F ( z ). Regarding Notation 9.2, we can infer then h G ( z ) G ( z ) i = − [ ¨ W s ( z )] h F ( z ) I q i .Using Lemma 9.10, we obtain h ¨ V s ( z ) i " G ( z ) G ( z ) = − h ¨ V s ( z ) ih ¨ W s ( z ) i " F ( z ) I q = ( β − z ) δ " P R ( s ) F ( z ) I q = ( β − z ) δ " F ( z ) I q . In view of z = β and δ >
0, we can conclude q ≥ rank h G ( z ) G ( z ) i ≥ rank h F ( z ) I q i = q , implyingrank h G ( z ) G ( z ) i = q . Let x := z − α and let y := β − z . Furthermore, let W := ¨ W s ( z ), let55 := [( xI q ) ⊕ I q ] W , and let W := [( yI q ) ⊕ I q ] W . As already mentioned above, we have − W h F ( z ) I q i = h G ( z ) G ( z ) i . Consequently, − W " F ( z ) I q = " xG ( z ) G ( z ) and − W " F ( z ) I q = " yG ( z ) G ( z ) . Taking additionally into account P R ( s ) F ( z ) = F ( z ), the application of Proposition 9.4 yields " xG ( z ) G ( z ) ∗ ˜ J q " xG ( z ) G ( z ) = δ " yxF ( z ) I q ∗ ˜ J q " yxF ( z ) I q − z ) s ! and " yG ( z ) G ( z ) ∗ ˜ J q " yG ( z ) G ( z ) = δ | y | " F ( z ) I q ∗ ˜ J q " F ( z ) I q − z )[ F ( z )] ∗ s † [ F ( z )] ! . Because of Remark A.36, we have " xG ( z ) G ( z ) ∗ ˜ J q " xG ( z ) G ( z ) = 2 Im([ G ( z )] ∗ [ xG ( z )]) = 2 Im(( z − α )[ G ( z )] ∗ [ G ( z )]) , " yG ( z ) G ( z ) ∗ ˜ J q " yG ( z ) G ( z ) = 2 Im([ G ( z )] ∗ [ yG ( z )]) = 2 Im(( β − z )[ G ( z )] ∗ [ G ( z )]) , and, furthermore, " yxF ( z ) I q ∗ ˜ J q " yxF ( z ) I q = 2 Im[ yxF ( z )] and " F ( z ) I q ∗ ˜ J q " F ( z ) I q = 2 Im[ F ( z )] . Now assume in addition z / ∈ R . Taken all together, we get then1Im z Im(( z − α )[ G ( z )] ∗ [ G ( z )]) = δ (cid:18) z Im[ yxF ( z )] − s (cid:19) (11.1)and 1Im z Im(( β − z )[ G ( z )] ∗ [ G ( z )]) = δ | y | (cid:18) z Im[ F ( z )] − [ F ( z )] ∗ s † [ F ( z )] (cid:19) . (11.2)Remark 4.22 provides us z Im[ yxF ( z )] < s . Taking additionally into account δ > z Im(( z − α )[ G ( z )] ∗ [ G ( z )]) ∈ C q × q < . More-over, in view of Theorem 4.9, we apply Lemma C.7 to F and obtain [ F ( z )] s † [ F ( z )] ∗ (Im z ) − Im[ F ( z )]. Because of Remark 4.8, we have (Im z ) − Im[ F ( z )] − [ F ( z )] s † [ F ( z )] ∗ = z Im[ F ( z )] − [ F ( z )] ∗ s † [ F ( z )]. Taking additionally into account δ > z Im(( β − z )[ G ( z )] ∗ [ G ( z )]) ∈ C q × q < . In view of thechoice of z in C \ [ α, β ] = C \ ([ α, β ] ∪ D ), we have thus shown that the conditions (II)–(IV)in Notation 7.5 are fulfilled with the set D for the pair [ P ; Q ] = [ G ; G ]. Consequently,[ G ; G ] ∈ PR q ( C \ [ α, β ]). Furthermore, P R ( s ) F = F is verified. According to Definition 9.1,then P R ( s ) G = G . By virtue of Lemma 8.3, hence [ G ; G ] ∈ ¨ P [ s ] follows.56 emma 11.2. Assume κ ≥ . Let ( s j ) κj =0 ∈ F < q,κ,α,β with F α,β -transform ( t j ) κ − j =0 and let F ∈R q [[ α, β ]; ( s j ) κj =0 ] . Further, let a be given via Notation 3.2. Then the F α,β ( a , s ) -transformof F belongs to R q [[ α, β ]; ( t j ) κ − j =0 ] .Proof. We first consider the case κ = ∞ . Let ρ := max {| α | , | β |} and let C ρ := { z ∈ C : | z | > ρ } .Obviously, C ρ ⊆ C \ [ α, β ] and 0 / ∈ C ρ . According to Proposition 3.9, the sequences ( a j ) ∞ j =0 and ( b j ) ∞ j =0 introduced in Notation 3.2 both belong to F < q, ∞ ,α,β . Because of Remark 5.10,the matrix-valued functions F a and F b given in Notation 4.19 fulfill F a ∈ R q [[ α, β ]; ( a j ) ∞ j =0 ]and F b ∈ R q [[ α, β ]; ( b j ) ∞ j =0 ]. Consequently, F b and F a are holomorphic in C \ [ α, β ] andProposition 5.8 yields, for all z ∈ C ρ , the series expansions F b ( z ) = − ∞ X j =0 z − ( j +1) b j and F a ( z ) = − ∞ X j =0 z − ( j +1) a j . In accordance with Definition 3.48, denote by ( g j ) ∞ j =0 the ( −∞ , β ]-modification of ( a j ) ∞ j =0 .Then g = − a and, in view of Notation 3.2, furthermore g j = βa j − − a j for all j ∈ N . Thematrix-valued functions Y, Z : C \ [ α, β ] → C q × q defined by Y ( z ) := − zF b ( z ) and Z ( z ) := − ( β − z ) F a ( z ), resp., are both holomorphic in C \ [ α, β ] with series expansions Y ( z ) = ∞ X n =0 z − n b n and Z ( z ) = ∞ X n =0 z − n g n for all z ∈ C ρ . Let R : C \ [ α, β ] → C q × q be defined by R ( z ) := [ Z ( z )] † . Observe thatthe R [ α, β ]-measure ¨ σ a of F a fulfills ¨ σ a ([ α, β ]) = a . Using Proposition 4.15, we obtain, for all z ∈ C \ [ α, β ], thus R ( Z ( z )) = R ( F a ( z )) = R ( a ) = R ( g ) and, analogously, N ( Z ( z )) = N ( g ).Therefore, we see from Proposition F.4 that the matrix-valued function R is holomorphic in C \ [ α, β ]. Denote by ( r j ) ∞ j =0 the reciprocal sequence associated to ( g j ) ∞ j =0 . The application ofLemma F.8 yields the series expansion R ( z ) = ∞ X n =0 z − n r n for all z ∈ C ρ . Denote by ( x j ) ∞ j =0 the Cauchy product of ( b j ) ∞ j =0 and ( r j ) ∞ j =0 . From Lemma F.7we see then that the function X := Y R is holomorphic in C \ [ α, β ] with series expansion X ( z ) = ∞ X n =0 z − n x n for all z ∈ C ρ . Observe that the R [ α, β ]-measure ¨ σ F of F fulfills ¨ σ F ([ α, β ]) = s . Denote by G the F α,β ( a , s )-transform of F . Regarding Notation 4.19 and Definition 10.1, we obtain G ( z ) = a s † [ F b ( z )][( β − z ) F a ( z )] † a = a s † [ F b ( z )][ − Z ( z )] † a = − a s † [ F b ( z )][ R ( z )] a for all z ∈ C \ [ α, β ]. In particular, by virtue of Remarks F.2 and F.1, thus G is holomorphicin C \ [ α, β ]. Let H : C \ [ α, β ] → C q × q be defined by H ( z ) := zG ( z ). We have H ( z ) = − za s † [ F b ( z )][ R ( z )] a = a s † [ Y ( z )][ R ( z )] a = a s † [ X ( z )] a z ∈ C \ [ α, β ]. From Remark F.6 we see then that the matrix-valued function H isholomorphic in C \ [ α, β ] with series expansion H ( z ) = ∞ X n =0 z − n ( a s † x n a )for all z ∈ C ρ . In view of Definition 3.51, consequently G ( z ) = 1 z H ( z ) = − ∞ X j =0 z − ( j +1) t j for all z ∈ C ρ follows. Due to Proposition 3.58, the sequence ( t j ) ∞ j =0 belongs to F < q, ∞ ,α,β . Since G is holomorphic in C \ [ α, β ], the application of Proposition 5.9 thus yields G ∈ R q [[ α, β ]; ( t j ) ∞ j =0 ],completing the proof in the case κ = ∞ .Now we consider the case κ < ∞ . Then m := κ belongs to N . Regarding Remark 4.11,denote by ˆ s j := R [ α,β ] ξ j ¨ σ F (d ξ ) for all j ∈ N the power moments of the R [ α, β ]-measure ¨ σ F of F . Then ¨ σ F ∈ M < q, ∞ [[ α, β ]; (ˆ s j ) ∞ j =0 , =], i. e., F ∈ R q [[ α, β ]; (ˆ s j ) ∞ j =0 ]. In particular, by virtue ofProposition 5.6, we therefore have (ˆ s j ) ∞ j =0 ∈ F < q, ∞ ,α,β . Denote by (ˆ t j ) ∞ j =0 the F α,β -transformof (ˆ s j ) ∞ j =0 . Let ˆ a := − α ˆ s + ˆ s and denote by G the F α,β (ˆ a , ˆ s )-transform of F . Since theassertion is already proved for κ = ∞ , we see that G belongs to R q [[ α, β ]; (ˆ t j ) ∞ j =0 ]. Observethat by assumption m ≥ s j = ˆ s j for all j ∈ Z ,m hold true. Hence, we have ˆ a = a and,because of Remark 3.52, furthermore ˆ t j = t j for all j ∈ Z ,m − . Consequently, G is exactly the F α,β ( a , s )-transform of F and belongs to R q [[ α, β ]; ( t j ) m − j =0 ].
12. On the elementary steps of the backward algorithm
This section can be considered as the analogue of the preceding one for the backward algorithm.More precisely, we will work out the elementary step of the backward algorithm by applyingthe transformation studied in Section 10.
Lemma 12.1.
Let ( s j ) j =0 ∈ F < q, ,α,β and let [ G ; G ] ∈ ¨ P [ s ] . Then the inverse F α,β ( s ) -transform of [ G ; G ] belongs to R q [[ α, β ]; ( s j ) j =0 ] .Proof. According to Notation 8.1, we have [ G ; G ] ∈ PR q ( C \ [ α, β ]). In particular, G and G are C q × q -valued functions, which are meromorphic in C \ [ α, β ]. Using the functions g, h : C \ [ α, β ] → C given via (9.9), we define by (9.10) two C q × q -valued functions F and F meromorphic in C \ [ α, β ]. From Remark 3.11 we get s ∈ C q × q < . By virtue of Proposition 9.11,thus det F does not vanish identically in C \ [ α, β ] and the inverse F α,β ( s )-transform F of[ G ; G ] admits the representation F = F F − , according to Definition 9.5.In a first step, we are now going to show that the pair [ F ; F ] belongs to PR q ( C \ [ α, β ]).Due to Proposition 7.9, the set A := P ( G ) ∪ P ( G ) ∪ ¨ E ([ G ; G ]) is a discrete subset D of C \ [ α, β ], satisfying the conditions (I)–(IV) in Notation 7.5 for the pair [ P ; Q ] = [ G ; G ]. Inview of (9.10), we have P ( F ) ⊆ P ( G ) ∪ P ( G ) and P ( F ) ⊆ P ( G ) ∪ P ( G ). Consequently, P ( F ) ∪ P ( F ) ⊆ A , i. e., condition (I) in Notation 7.5 is fulfilled with the set D = A forthe pair [ P ; Q ] = [ F ; F ]. Consider an arbitrary z ∈ C \ ([ α, β ] ∪ A ). Regarding s ∈ C q × q < ,Proposition 9.11 yields det F ( z ) = 0 and F ( z ) = [ F ( z )][ F ( z )] − . (12.1)58n particular, rank h F ( z ) F ( z ) i = q . Let x := z − α and let y := β − z . Furthermore, let V := ¨ V s ( z ),let V := [( xI q ) ⊕ I q ] V , and let V := [( yI q ) ⊕ I q ] V . According to Notation 9.6, we have V h G ( z ) G ( z ) i = h F ( z ) F ( z ) i . Hence, V " G ( z ) G ( z ) = " xF ( z ) F ( z ) and V " G ( z ) G ( z ) = " yF ( z ) F ( z ) . From Lemma 8.3 we see P R ( s ) G = G . Using Proposition 9.9, we thus can infer " xF ( z ) F ( z ) ∗ ˜ J q " xF ( z ) F ( z ) = | y | " xG ( z ) G ( z ) ∗ ˜ J q " xG ( z ) G ( z ) + | x | " yG ( z ) G ( z ) ∗ ˜ J q " yG ( z ) G ( z ) + 2 δ Im( z )[ G ( z )] ∗ s [ G ( z )]and " yF ( z ) F ( z ) ∗ ˜ J q " yF ( z ) F ( z ) = δ | y | " G ( z ) G ( z ) ∗ ˜ J q " G ( z ) G ( z ) + 2 Im( z )[ G ( z )] ∗ s † [ G ( z )] ! . Because of Remark A.36, we see that " ξF ( z ) F ( z ) ∗ ˜ J q " ξF ( z ) F ( z ) = 2 Im([ F ( z )] ∗ [ ξF ( z )]) = 2 Im( ξ [ F ( z )] ∗ [ F ( z )])and " ξG ( z ) G ( z ) ∗ ˜ J q " ξG ( z ) G ( z ) = 2 Im([ G ( z )] ∗ [ ξG ( z )]) = 2 Im( ξ [ G ( z )] ∗ [ G ( z )])hold true for each ξ ∈ { x, y } and that " G ( z ) G ( z ) ∗ ˜ J q " G ( z ) G ( z ) = 2 Im([ G ( z )] ∗ [ G ( z )])is valid. Now assume in addition z / ∈ R . Taken all together, we obtain then1Im z Im( x [ F ( z )] ∗ [ F ( z )]) = | y | (cid:20) z Im( x [ G ( z )] ∗ [ G ( z )]) (cid:21) + | x | (cid:20) z Im( y [ G ( z )] ∗ [ G ( z )]) (cid:21) + δ [ G ( z )] ∗ s [ G ( z )] (12.2)and 1Im z Im( y [ F ( z )] ∗ [ F ( z )]) = δ | y | (cid:20) z Im([ G ( z )] ∗ [ G ( z )]) + [ G ( z )] ∗ s † [ G ( z )] (cid:21) . (12.3)Since the conditions (III) and (IV) in Notation 7.5 are satisfied with the set D = A for thepair [ P ; Q ] = [ G ; G ], we have1Im z Im( x [ G ( z )] ∗ [ G ( z )]) ∈ C q × q < and 1Im z Im( y [ G ( z )] ∗ [ G ( z )]) ∈ C q × q < . (12.4)59aking into account (12.4), δ >
0, and s ∈ C q × q < , we use Remarks A.24 and A.25 to in-fer from (12.2) that z Im( x [ F ( z )] ∗ [ F ( z )]) ∈ C q × q < . By virtue of Lemma 7.8, the ma-trix z Im([ G ( z )] ∗ [ G ( z )]) is non-negative Hermitian. Because of Remark A.16, we have s † ∈ C q × q < . Regarding additionally δ >
0, from Remarks A.24 and A.25 and (12.3) we concludesimilarly z Im( y [ F ( z )] ∗ [ F ( z )]) ∈ C q × q < . In view of the choice of z ∈ C \ ([ α, β ] ∪ A ), we thushave verified that conditions (II)–(IV) in Notation 7.5 are fulfilled with the set D = A for thepair [ P ; Q ] = [ F ; F ]. Consequently, [ F ; F ] ∈ PR q ( C \ [ α, β ]). Therefore, the application ofLemma 7.13 to the pair [ P ; Q ] = [ F ; F ] yields F = F F − ∈ R q ( C \ [ α, β ]) . (12.5)In a second step, we are now going to show that ¨ σ F ([ α, β ]) = s . First we verify¨ σ F ([ α, β ]) < s . (12.6)Let W := ¨ W s ( z ). Because of Lemma 9.10 and P R ( s ) G = G , we have W " F ( z ) F ( z ) = W V " G ( z ) G ( z ) = − yδ " P R ( s ) G ( z ) G ( z ) = − yδ " G ( z ) G ( z ) . (12.7)Setting W := [( xI q ) ⊕ I q ] W , hence W h F ( z ) F ( z ) i = − yδ h xG ( z ) G ( z ) i follows. In view of (9.10), wehave P R ( s ) F = F . Thus, using Proposition 9.4, we can conclude then | y | δ " xG ( z ) G ( z ) ∗ ˜ J q " xG ( z ) G ( z ) = δ " yxF ( z ) F ( z ) ∗ ˜ J q " yxF ( z ) F ( z ) − z )[ F ( z )] ∗ s [ F ( z )] ! . Remark A.36 yields h xG ( z ) G ( z ) i ∗ ˜ J q h xG ( z ) G ( z ) i = 2 Im( x [ G ( z )] ∗ [ G ( z )]). Consequently, | y | δ (cid:20) z Im( x [ G ( z )] ∗ [ G ( z )]) (cid:21) = 12 Im z " yxF ( z ) F ( z ) ∗ ˜ J q " yxF ( z ) F ( z ) − [ F ( z )] ∗ s [ F ( z )] . Taking into account δ > J q and s are Hermitianmatrices, we can then use Remarks A.24 and A.25 to conclude[ F ( z )] ∗ s [ F ( z )]
12 Im z " yxF ( z ) F ( z ) ∗ ˜ J q " yxF ( z ) F ( z ) . In view of (12.1) and Remarks A.25 and A.36, hence s
12 Im z " yxF ( z ) I q ∗ ˜ J q " yxF ( z ) I q = 1Im z Im[ yxF ( z )] = 1Im z Im[( β − z )( z − α ) F ( z )]follows. Since A is a discrete set, we in particular inferlim η →∞ η Im[( β − i η )(i η − α ) F (i η )] < s . (12.8)60or all η >
0, we have η − ( β − i η )(i η − α ) = ( βη − − i)(i − αη − ) and thereforelim η →∞ η − ( β − i η )(i η − α ) = 1. Regarding (12.5) and Theorem 4.9, we can apply Lemma C.4to F and obtain lim η →∞ i ηF (i η ) = − ¨ σ F ([ α, β ]). Consequently, − ¨ σ F ([ α, β ]) = (cid:20) lim η →∞ η ( β − i η )(i η − α ) (cid:21)(cid:20) lim η →∞ i ηF (i η ) (cid:21) = lim η →∞ i η [( β − i η )(i η − α ) F (i η )] . Because of ¨ σ F ([ α, β ]) ∈ C q × q H , we obtain from Remark A.2 then¨ σ F ([ α, β ]) = − Re (cid:18) lim η →∞ i η [( β − i η )(i η − α ) F (i η )] (cid:19) = lim η →∞ η Im[( β − i η )(i η − α ) F (i η )] . In combination with (12.8), this implies (12.6).Conversely, we now verify ¨ σ F ([ α, β ]) s . Because of (12.5) and Proposition 4.15, we have R ( F ( z )) = R (¨ σ F ([ α, β ])). Using Remark A.8 and (12.1), we infer R ( F ( z )) = R ( F ( z )). Inview of P R ( s ) F = F , we have R ( F ( z )) ⊆ R ( s ). Consequently, R (¨ σ F ([ α, β ])) ⊆ R ( s ). Let W := [( yI q ) ⊕ I q ] W . In view of (12.7), then W (cid:2) F ( z ) F ( z ) (cid:3) = − yδ (cid:2) yG ( z ) G ( z ) (cid:3) . Taking additionallyinto account P R ( s ) F = F , from Proposition 9.4 we get | y | δ " yG ( z ) G ( z ) ∗ ˜ J q " yG ( z ) G ( z ) = δ | y | " F ( z ) F ( z ) ∗ ˜ J q " F ( z ) F ( z ) − z )[ F ( z )] ∗ s † [ F ( z )] ! . Remark A.36 yields h yG ( z ) G ( z ) i ∗ ˜ J q h yG ( z ) G ( z ) i = 2 Im( y [ G ( z )] ∗ [ G ( z )]). Consequently, δ (cid:20) z Im( y [ G ( z )] ∗ [ G ( z )]) (cid:21) = 12 Im z " F ( z ) F ( z ) ∗ ˜ J q " F ( z ) F ( z ) − [ F ( z )] ∗ s † [ F ( z )] . Regarding δ > s is Hermitian, Remark A.14 shows that s † is Hermitian as well. Taking additionally into account that ˜ J q is Hermitian, we can useRemarks A.24 and A.25 to conclude[ F ( z )] ∗ s † [ F ( z )]
12 Im z " F ( z ) F ( z ) ∗ ˜ J q " F ( z ) F ( z ) . Because of (12.1), the application of Remarks A.25 and A.36 thus yields[ F ( z )] ∗ s † [ F ( z )]
12 Im z " F ( z ) I q ∗ ˜ J q " F ( z ) I q = 1Im z Im F ( z ) . Regarding that the set A is discrete and that, according to (12.5), the function F is holomorphic,we can apply a continuity argument to show that [ F ( ζ )] ∗ s † [ F ( ζ )] (Im ζ ) − Im F ( ζ ) holds truefor all ζ ∈ Π + . In view of (12.5) and Theorem 4.9, we can apply Lemma C.8 to F . Takingadditionally into account s ∈ C q × q < and R (¨ σ F ([ α, β ])) ⊆ R ( s ), then ¨ σ F ([ α, β ]) s followsby Lemma C.8. In combination with (12.6), this implies ¨ σ F ([ α, β ]) = s . Because of (12.5),thus F belongs to R q [[ α, β ]; ( s j ) j =0 ]. Lemma 12.2.
Assume κ ≥ . Let ( s j ) κj =0 ∈ F < q,κ,α,β with F α,β -transform ( t j ) κ − j =0 and let G ∈R q [[ α, β ]; ( t j ) κ − j =0 ] . Then the inverse F α,β ( a , s ) -transform of G belongs to R q [[ α, β ]; ( s j ) κj =0 ] . roof. Because of Proposition 3.58, we have ( t j ) κ − j =0 ∈ F < q,κ − ,α,β . First we consider the case κ = ∞ . By virtue of Proposition 5.8, the set R q [[ α, β ]; ( s j ) ∞ j =0 ] consists of exactly one element,say F . According to Lemma 11.2, the F α,β ( a , s )-transform of F belongs to R q [[ α, β ]; ( t j ) ∞ j =0 ].Since, due to Proposition 5.8, the set R q [[ α, β ]; ( t j ) ∞ j =0 ] consists of exactly one element, weconclude that G coincides with the F α,β ( a , s )-transform of F . Using Remark 3.11, we easilyinfer a ∈ C q × q H and s ∈ C q × q < . Due to Remark 3.12, we have R ( a ) ⊆ R ( s ). Consider nowan arbitrary z ∈ C \ [ α, β ]. Observe that the R [ α, β ]-measure ¨ σ F of F fulfills ¨ σ F ([ α, β ]) = s .Taking into account Proposition 4.15, we obtain thus R ( F ( z )) = R ( s ) and N ( F ( z )) = N ( s ).Because of Remark 5.10, the function F a given in Notation 4.19 belongs to R q [[ α, β ]; ( a j ) ∞ j =0 ].Hence, we analogously get R ( F a ( z )) = R ( a ) and N ( F a ( z )) = N ( a ). In a similar way, wecan conclude R ( G ( z )) = R ( t ). Regarding Definition 3.51, we see furthermore R ( t ) ⊆ R ( a ).Consequently, R ( G ( z )) ⊆ R ( a ). Thus, we can apply Lemma 10.13 to the function F andits F α,β ( a , s )-transform G and obtain with the inverse F α,β ( a , s )-transform H of G then H ( z ) = F ( z ). Hence, H = F , implying H ∈ R q [[ α, β ]; ( s j ) ∞ j =0 ].Now we consider the case κ < ∞ . Then m := κ belongs to N . Regarding Remark 4.11, de-note by ˆ t j := R [ α,β ] ξ j ¨ σ G (d ξ ) for all j ∈ N the power moments of the R [ α, β ]-measure ¨ σ G of G .Then ¨ σ G ∈ M < q, ∞ [[ α, β ]; (ˆ t j ) ∞ j =0 , =], i. e., G ∈ R q [[ α, β ]; (ˆ t j ) ∞ j =0 ]. By virtue of Proposition 5.6,we have in particular (ˆ t j ) ∞ j =0 ∈ F < q, ∞ ,α,β . We are now going to construct the F α,β -parametersequence of a sequence from F < q, ∞ ,α,β with F α,β -transform (ˆ t j ) ∞ j =0 : Due to Theorem 3.41, the F α,β -parameter sequence (ˆ g j ) ∞ j =0 of (ˆ t j ) ∞ j =0 belongs to the class C < q, ∞ ,δ introduced in Nota-tion 3.40. Since G belongs to R q [[ α, β ]; ( t j ) m − j =0 ], we have ˆ t j = t j for all j ∈ Z ,m − . Denote by( g j ) m − j =0 the F α,β -parameter sequence of ( t j ) m − j =0 . Because of Remark 3.37, then ˆ g j = g j forall j ∈ Z , m − . Denote by ( f j ) mj =0 the F α,β -parameter sequence of ( s j ) mj =0 . Regarding δ > f j ) ∞ j =0 be given byˆ f j := ( f j , if j ≤ mδ − ˆ g j − , if j ≥ m + 1 . Theorem 3.41 yields ( f j ) mj =0 ∈ C < q,m,δ . Taking additionally into account δ > m ≥
1, wecan conclude then that the sequence (ˆ f j ) ∞ j =0 is a sequence of non-negative Hermitian matricesfulfilling the relations δ ˆ f = δ f = f + f = ˆ f + ˆ f and δ (ˆ f k − ⊤−⊥ ˆ f k ) = δ ( f k − ⊤−⊥ f k ) = f k +1 + f k +2 = ˆ f k +1 + ˆ f k +2 for all k ∈ N with k ≤ m −
1, and, regarding Remark A.17, furthermore δ (ˆ f k − ⊤−⊥ ˆ f k ) = δ h ( δ − ˆ g k − ) ⊤−⊥ ( δ − ˆ g k − ) i = ˆ g k − ⊤−⊥ ˆ g k − = δ − (ˆ g k − + ˆ g k ) = ˆ f k +1 +ˆ f k +2 for all k ∈ N with k ≥ m + 1. In view of Corollary 3.54, in addition g = δ ( f ⊤−⊥ f ) and g j = δ f j +2 hold true for all j ∈ Z , m − . In the case m = 1, we therefore have δ ( f m − ⊤−⊥ f m ) = δ ( f ⊤−⊥ f ) = g = ˆ g = δ − (ˆ g + ˆ g ) = δ − (ˆ g m − + ˆ g m ) , whereas, because of Remark A.17, in the case m ≥ δ ( f m − ⊤−⊥ f m ) = ( δ f m − ) ⊤−⊥ ( δ f m ) = g m − ⊤−⊥ g m − = ˆ g m − ⊤−⊥ ˆ g m − = δ − (ˆ g m − + ˆ g m )62ollows. Consequently, δ (ˆ f m − ⊤−⊥ ˆ f m ) = ˆ f m +1 + ˆ f m +2 . Hence, the sequence (ˆ f j ) ∞ j =0 belongsto C < q, ∞ ,δ . According to Theorem 3.41, then there exists a sequence (ˆ s j ) ∞ j =0 from F < q, ∞ ,α,β with F α,β -parameter sequence (ˆ f j ) ∞ j =0 . Denote by (˜ t j ) ∞ j =0 the F α,β -transform of (ˆ s j ) ∞ j =0 and by(˜ g j ) ∞ j =0 the F α,β -parameter sequence of (˜ t j ) ∞ j =0 . Using Corollary 3.54, we infer˜ g = δ (ˆ f ⊤−⊥ ˆ f ) = δ ( f ⊤−⊥ f ) = g = ˆ g and ˜ g j = δ ˆ f j +2 = δ f j +2 = g j = ˆ g j for all j ∈ N with j ≤ m − g j = δ ˆ f j +2 = ˆ g j for all j ∈ N with j ≥ m −
1. Consequently, the F α,β -parameter sequence (ˆ g j ) ∞ j =0 of (ˆ t j ) ∞ j =0 coincides with the F α,β -parameter sequence (˜ g j ) ∞ j =0 of (˜ t j ) ∞ j =0 . Since Proposition 3.58 yields (˜ t j ) ∞ j =0 ∈ F < q, ∞ ,α,β , wecan conclude from Theorem 3.41 that the sequences (ˆ t j ) ∞ j =0 and (˜ t j ) ∞ j =0 coincide. In particular,(ˆ t j ) ∞ j =0 is the F α,β -transform of (ˆ s j ) ∞ j =0 . The application of the already for κ = ∞ proved as-sertion of the present lemma yields with ˆ a := − α ˆ s + ˆ s for the inverse F α,β (ˆ a , ˆ s )-transform H of G thus H ∈ R q [[ α, β ]; (ˆ s j ) ∞ j =0 ]. In view of Proposition 3.8 and Remark 3.37, the sequence(ˆ s j ) mj =0 belongs to F < q,m,α,β and its F α,β -parameter sequence is exactly (ˆ f j ) mj =0 . Consequently,the F α,β -parameter sequences of (ˆ s j ) mj =0 and ( s j ) mj =0 coincide as well. By virtue of Theo-rem 3.41, then the sequences (ˆ s j ) mj =0 and ( s j ) mj =0 coincide. Hence, H is exactly the inverse F α,β ( a , s )-transform of G and belongs to R q [[ α, β ]; ( s j ) mj =0 ].
13. Parametrization of the set of all solutions
We are now going to iterate the F α,β ( M )-transformation introduced in Definition 9.1 withthe F α,β ( A, M )-transformation introduced in Definition 10.1. To that end, we use the k -th F α,β -transform ( s { k } j ) κ − kj =0 of a sequence ( s j ) κj =0 constructed in Definition 3.55 and, in addition,the sequence ( a { k } j ) κ − − kj =0 given by a { k } j := − αs { k } j + s { k } j +1 , i. e., the sequence built from ( s { k } j ) κ − kj =0 according to Notation 3.2: Definition 13.1.
Let G be a non-empty subset of C , let F : G → C p × q be a matrix-valuedfunction, and let ( s j ) κj =0 be a sequence of complex p × q matrices. Let ¨ G ( F, ( s j ) κj =0 ) := F .Recursively, for all k ∈ Z ,κ , denote by ¨ G k ( F, ( s j ) κj =0 ) the F α,β ( a { k − } , s { k − } )-transform of¨ G k − ( F, ( s j ) κj =0 ). In view of Definition 9.1, for all k ∈ Z ,κ , denote by P ¨ G k ( F, ( s j ) κj =0 ) the F α,β ( s { k } )-transformed pair of ¨ G k ( F, ( s j ) κj =0 ). Then, for all m ∈ Z ,κ , we call P ¨ G m ( F, ( s j ) κj =0 )the m -th F α,β -transformed pair of F with respect to ( s j ) κj =0 and we call ¨ G m ( F, ( s j ) κj =0 ) the m -th F α,β -transform of F with respect to ( s j ) κj =0 . Remark . The pair P ¨ G ( F, ( s j ) κj =0 ) is exactly the F α,β ( s )-transformed pair of F . If κ ≥ G ( F, ( s j ) κj =0 ) is exactly the F α,β ( a , s )-transform of F .Regarding Proposition 9.11, we will use the following mappings: Notation . For each matrix M ∈ C q × q < , let the mapping ¨ F M be defined on the class ¨ P [ M ] by¨ F M ([ G ; G ]) := F , where F is the inverse F α,β ( M )-transform of [ G ; G ]. Furthermore, giventwo matrices A, M ∈ C p × q , let the mapping ¨ F A,M be defined on the set of all matrix-valuedfunctions G : C \ [ α, β ] → C p × q by ¨ F A,M ( G ) := F , where F is the inverse F α,β ( A, M )-transformof G . 63 roposition 13.4. Let ( s j ) j =0 ∈ F < q, ,α,β . Then ψ : h ¨ P [ s ] i → R q [[ α, β ]; ( s j ) j =0 ] de-fined by ψ ( h [ G ; G ] i ) := ¨ F s ([ G ; G ]) is a bijection with inverse ψ − given by ψ − ( F ) = h P ¨ G ( F, ( s j ) j =0 ) i .Proof. Consider arbitrary [ G ; G ] ∈ ¨ P [ s ] and F ∈ R q [[ α, β ]; ( s j ) j =0 ]. Due to Remark 3.11,we have s ∈ C q × q < . According to Corollary 9.12, then ψ ( h [ G ; G ] i ) is independent of theconcrete representative of the equivalence class h [ G ; G ] i . Since, because of Lemma 12.1, fur-thermore ¨ F s ([ G ; G ]) belongs to R q [[ α, β ]; ( s j ) j =0 ], the mapping ψ is well defined. RegardingRemark 13.2, we obtain from Lemma 11.1 moreover P ¨ G ( F, ( s j ) j =0 ) ∈ ¨ P [ s ]. Consequently,the mapping χ : R q [[ α, β ]; ( s j ) j =0 ] → h ¨ P [ s ] i defined by χ ( S ) := h P ¨ G ( S, ( s j ) j =0 ) i is well de-fined as well. Using Lemma 9.13 and Remark 13.2, we conclude ( χ ◦ ψ )( h [ G ; G ] i ) = h [ G ; G ] i .Because of Remark 5.7 and Lemma C.5, we get R ( F ( z )) = R ( s ) for all z ∈ C \ [ α, β ]. There-fore, P R ( s ) F = F . Taking additionally into account Remark 13.2 and P ¨ G ( F, ( s j ) j =0 ) ∈ ¨ P [ s ],Lemma 9.14 then yields ( ψ ◦ χ )( F ) = F . Consequently, ψ is a bijection with inverse χ . Proposition 13.5.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β with F α,β -transform ( t j ) m − j =0 . Then ψ : R q [[ α, β ]; ( t j ) m − j =0 ] → R q [[ α, β ]; ( s j ) mj =0 ] defined by ψ ( G ) := ¨ F a ,s ( G ) is a bijection withinverse ψ − given by ψ − ( F ) = ¨ G ( F, ( s j ) mj =0 ) .Proof. In view of Lemma 12.2, the mapping ψ is well defined. Regarding Remark 13.2, wesee from Lemma 11.2 that the mapping χ : R q [[ α, β ]; ( s j ) mj =0 ] → R q [[ α, β ]; ( t j ) m − j =0 ] given by χ ( F ) := ¨ G ( F, ( s j ) mj =0 ) is also well defined. Using Remark 3.11, we easily infer a ∈ C q × q H and s ∈ C q × q < . Because of Remark 3.12, we have R ( a ) ⊆ R ( s ). Consider now an arbi-trary G ∈ R q [[ α, β ]; ( t j ) m − j =0 ]. Then G ∈ R q ( C \ [ α, β ]). Taking into account Remark 5.7, weconclude from Lemma C.5 furthermore R ( G ( z )) = R ( t ) for all z ∈ C \ [ α, β ]. RegardingDefinition 3.51, we thus obtain R ( G ( z )) ⊆ R ( a ) for all z ∈ C \ [ α, β ]. By construction, ψ ( G )is the inverse F α,β ( a , s )-transform of G . Denote by H the F α,β ( a , s )-transform of ψ ( G ).In view of Remark 13.2, then H = χ ( ψ ( G )). Using Lemma 10.12, hence H = G follows. Con-sequently, ( χ ◦ ψ )( G ) = G . Now we consider an arbitrary F ∈ R q [[ α, β ]; ( s j ) mj =0 ]. By virtueof Lemma 11.2, the F α,β ( a , s )-transform χ ( F ) of F then belongs to R q [[ α, β ]; ( t j ) m − j =0 ]. Asabove, we thus have χ ( F ) ∈ R q ( C \ [ α, β ]) and R ([ χ ( F )]( z )) ⊆ R ( a ) for all z ∈ C \ [ α, β ].Observe that the R [ α, β ]-measure ¨ σ F of F satisfies ¨ σ F ([ α, β ]) = s . Using Proposition 4.15,we get R ( F ( z )) = R ( s ) and N ( F ( z )) = N ( s ) for all z ∈ C \ [ α, β ]. Because of Remark 5.10,the function F a given in Notation 4.19 belongs to R q [[ α, β ]; ( a j ) m − j =0 ]. Therefore, we obtainanalogously R ( F a ( z )) = R ( a ) and N ( F a ( z )) = N ( a ) for all z ∈ C \ [ α, β ]. Hence, we canapply Lemma 10.13 to the function F and its F α,β ( a , s )-transform χ ( F ) and obtain with theinverse F α,β ( a , s )-transform H = ψ ( χ ( F )) of χ ( F ) then H ( z ) = F ( z ) for all z ∈ C \ [ α, β ].Thus, ( ψ ◦ χ )( F ) = F . Consequently, ψ is bijective with inverse χ .The combination of Propositions 13.4 and 13.5 now yields a first parametrization of thesolution set of the matricial Hausdorff moment problem MP [[ α, β ]; ( s j ) mj =0 , =], where, however,the set of parameters still depends on the given data. Theorem 13.6.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Let ψ m : h ¨ P [ s { m } ] i →R q [[ α, β ]; ( s { m } j ) j =0 ] be defined by ψ m ( h [ G ; G ] i ) := ¨ F s { m } ([ G ; G ]) . In the case m ≥ let, for all k ∈ Z ,m − , furthermore ψ k : R q [[ α, β ]; ( s { k +1 } j ) m − k − j =0 ] → R q [[ α, β ]; ( s { k } j ) m − kj =0 ]64 e given by ψ k ( G ) := ¨ F a { k } ,s { k } ( G ) . Then Ψ m : h ¨ P [ s { m } ] i → R q [[ α, β ]; ( s j ) mj =0 ] defined by Ψ m ( h [ G ; G ] i ) := ( ψ ◦ ψ ◦ · · · ◦ ψ m )( h [ G ; G ] i ) is a bijection with inverse Ψ − m given by Ψ − m ( F ) = h P ¨ G m ( F, ( s j ) mj =0 ) i .Proof. Because of Proposition 3.58, we have ( s { k } j ) m − kj =0 ∈ F < q,m − k,α,β for all k ∈ Z ,m . Ac-cording to Proposition 13.4, then ψ m is a bijection with inverse ψ − m given by ψ − m ( F ) = h P ¨ G ( F, ( s { m } j ) j =0 ) i . Regarding Definition 3.55, we infer in the case m ≥ k ∈ Z ,m − from Proposition 13.5 that ψ k is a bijection with inverse ψ − k given by ψ − k ( F ) =¨ G ( F, ( s { k } j ) m − kj =0 ). In view of Remark 13.2, we see, for all F ∈ R q [[ α, β ]; ( s { m } j ) j =0 ], that ψ − m ( F )is exactly the equivalence class of the F α,β ( s { m } )-transformed pair of F . In the case m ≥
1, forall k ∈ Z ,m − and all F ∈ R q [[ α, β ]; ( s { k } j ) m − kj =0 ], furthermore ψ − k ( F ) coincides with the equiv-alence class of the F α,β ( a { k } , s { k } )-transform of F . Regarding Definition 13.1, we obtain in thecase m ≥
1, for all F ∈ R q [[ α, β ]; ( s j ) mj =0 ], the equation ( ψ − m − ◦ · · · ◦ ψ − )( F ) = ¨ G m ( F, ( s j ) mj =0 )and hence ( ψ − m ◦ ψ − m − ◦ · · · ◦ ψ − )( F ) = h P ¨ G m ( F, ( s j ) mj =0 ) i , implying that Ψ m is a bijectionwith inverse Ψ − m given by Ψ − m ( F ) = h P ¨ G m ( F, ( s j ) mj =0 ) i .We end this section by mentioning a relation between the k -th F α,β -transform given inDefinition 13.1 of a matrix-valued function with respect to a sequence of matrices and the k -th R [ α, β ]-Schur transform introduced in Definition 6.8. We start with the case k = 1: Lemma 13.7.
Suppose κ ≥ . Let ( s j ) κj =0 ∈ F < q,κ,α,β and let F ∈ R q [[ α, β ]; ( s j ) κj =0 ] with F α,β ( a , s ) -transform G and first R [ α, β ] -Schur transform F { } . Then G = F { } .Proof. By assumption we have F ∈ R q ( C \ [ α, β ]) with R [ α, β ]-measure σ := ¨ σ F belongingto M < q,κ [[ α, β ]; ( s j ) κj =0 , =]. Observe that σ ∈ M < q, ∞ ([ α, β ]) according to Remark 4.11. Setting s j := R [ α,β ] x j σ (d x ) for all j ∈ Z κ +1 , ∞ , we have then σ ∈ M < q, ∞ [[ α, β ]; ( s j ) ∞ j =0 , =] and hence F ∈ R q [[ α, β ]; ( s j ) ∞ j =0 ]. In particular, Theorem 3.5 shows ( s j ) κj =0 ∈ F < q, ∞ ,α,β . Regarding κ ≥ a = − αs + s is not affected by the above extension of the sequence ( s j ) κj =0 . Thus,we can apply Lemma 11.2 to obtain G ∈ R q [[ α, β ]; ( t j ) ∞ j =0 ], where ( t j ) ∞ j =0 is the F α,β -transformof ( s j ) ∞ j =0 . Consequently, we have G ∈ R q ( C \ [ α, β ]) with R [ α, β ]-measure ν := ¨ σ G belongingto M < q, ∞ [[ α, β ]; ( t j ) ∞ j =0 , =]. By virtue of Proposition 5.5, in particular G = ¨ S ν . According toDefinition 6.5, the first M [ α, β ]-transform µ := σ { } of σ belongs to M < q, ∞ [[ α, β ]; ( t j ) ∞ j =0 , =]as well. Due to Proposition 3.6 and Theorem 3.5, the set M < q, ∞ [[ α, β ]; ( t j ) ∞ j =0 , =] consists ofat most one element. Hence, ν = µ follows. Since F { } = ¨ S µ by Definition 6.8, we infer then G = ¨ S ν = ¨ S µ = F { } . Proposition 13.8.
Let ( s j ) κj =0 ∈ F < q,κ,α,β , let k ∈ Z ,κ , and let F ∈ R q [[ α, β ]; ( s j ) κj =0 ] with k -th F α,β -transform ¨ G k ( F, ( s j ) κj =0 ) of F with respect to ( s j ) κj =0 and k -th R [ α, β ] -Schur trans-form F { k } of F . Then ¨ G k ( F, ( s j ) κj =0 ) = F { k } .Proof. We use mathematical induction. According to Definition 13.1, we have ¨ G ( F, ( s j ) κj =0 ) = F , whereas Remark 6.10 shows F { } = F . Hence, the assertion holds true for k = 0. Nowassume that κ ≥ G k − ( F, ( s j ) κj =0 ) = F { k − } is valid for some k ∈ Z ,κ . Set-ting T := F { k − } and t j := s { k − } j for all j ∈ Z ,κ − ( k − , then ¨ G k ( F, ( s j ) κj =0 ) is, by Defini-tion 13.1, exactly the F α,β ( − αt + t , t )-transform of T . Furthermore, Remark 6.11 yields65 ∈ R q [[ α, β ]; ( t j ) κ − ( k − j =0 ]. From Proposition 3.58 we infer ( t j ) κ − ( k − j =0 ∈ F < q,κ − ( k − ,α,β . Tak-ing additionally into account κ − ( k − ≥
1, we can apply Lemma 13.7 to the sequence( t j ) κ − ( k − j =0 and the function T to see that the F α,β ( − αt + t , t )-transform of T coincides withthe first R [ α, β ]-Schur transform of T , i. e., ¨ G k ( F, ( s j ) κj =0 ) = T { } . Since Remark 6.10 provides F { k } = T { } , the proof is complete.In view of Lemma 13.7, a more explicit description of the Schur–Nevanlinna type algorithmfor the class R q ( C \ [ α, β ]) considered in Section 6 can be given by means of matricial linearfractional transformations. For the sake of simplicity, we illustrate this for the scalar case q = 1, where this amounts to a scalar linear fractional transformation or a continued fractionexpansion of functions belonging to R ( C \ [ α, β ]). These considerations are along the lines ofthe classical results by Schur [50, 51] for the class S × ( D ) of holomorphic functions mappingthe open unit disc D := { z ∈ C : | z | < } into the closed unit disc D (cf. Notation F.9) and byNevanlinna [45] for the class R , (Π + ) introduced in Notation 4.2:Let f ∈ R ( C \ [ α, β ]) with R [ α, β ]-measure ¨ σ f . Then f ∈ R [[ α, β ]; ( s j ) ∞ j =0 ] with thesequence ( s j ) ∞ j =0 of power moments s j := R [ α,β ] x j ¨ σ f (d x ) associated with ¨ σ f belonging to F < , ∞ ,α,β , by virtue of Proposition 3.7. We see from Lemma 13.7 that the first R [ α, β ]-Schurtransform f { } of f is exactly the F α,β ( a , s )-transform of f . Consequently, Lemma 11.2yields f { } ∈ R [[ α, β ]; ( s { } j ) ∞ j =0 ] with the F α,β -transform ( s { } j ) ∞ j =0 of ( s j ) ∞ j =0 belonging to F < , ∞ ,α,β , by virtue of Proposition 3.58. Hence, we can conclude from Proposition 5.8 that theexpansions f ( z ) = − s z − s z − s z − · · · and f { } ( z ) = − s { } z − s { } z − s { } z − · · · are valid for all z ∈ C with | z | > max {| α | , | β |} . Recall that a = − αs + s and b = βs − s ,according to Notation 3.2. Lemma 3.11 yields s ≥ a ≥
0. In what follows, we assume s >
0. Because of (3.7) and Corollary 3.60, we have then d = δs and s { } = d = − αβs + ( α + β ) s − s s = a b s . In view of δ = β − α >
0, thus d >
0. Regarding Remark A.13 and a + b = δs , we obtain,by virtue of Definition 3.42 and (3.11), hence e = f = s and e = f d = b δs = δs − a δs = 1 − a δs . Taken all together, we can infer by direct calculation s = e , a = δ e (1 − e ), b = δ e e , and s { } = δ e e (1 − e ). In view of s = ¨ σ f ([ α, β ]), we can conclude from Remark 5.10 that thefunction f a : C \ [ α, β ] → C given, according to Notation 4.19, by f a ( z ) := ( z − α ) f ( z ) + s belongs to R ( C \ [ α, β ]) with R [ α, β ]-measure ¨ σ a fulfilling ¨ σ a ([ α, β ]) = a . If a = 0, thenProposition 4.15 yields f a ( z ) = 0, i. e., f ( z ) = ( α − z ) − s for all z ∈ C \ [ α, β ], implying¨ σ f = s δ α , by virtue of Proposition 5.5, where δ α is the Dirac measure on ([ α, β ] , B [ α,β ] ) withunit mass at α . Now assume a >
0. Proposition 4.15 yields, for all z ∈ C \ [ α, β ], then f a ( z ) = 0. In view of Definition 10.1 and Remark A.13, we thus can infer the representation f { } ( z ) = a /s β − z · ( β − z ) f ( z ) − s ( z − α ) f ( z ) + s = δ e (1 − e ) β − z · ( β − z ) f ( z ) − e ( z − α ) f ( z ) + e (13.1)66or all z ∈ C \ [ α, β ]. As seen above, we have f { } ∈ R q ( C \ [ α, β ]). Taking additionallyinto account f a ( z ) = 0 for all z ∈ C \ [ α, β ] and the assumptions s > a >
0, theconditions of Lemma 10.13 are fulfilled. So its application shows that f coincides with theinverse F α,β ( a , s )-transform of f { } . Observe that Q s = 0 and Q a = 0 by (9.2) and theassumptions s > a >
0. From Lemma 10.8 we can conclude for all z ∈ C \ [ α, β ] then a /s − ( z − α ) f { } ( z ) = 0 and the representation f ( z ) = s β − z · a /s + ( β − z ) f { } ( z ) a /s − ( z − α ) f { } ( z ) = e β − z · δ e (1 − e ) + ( β − z ) f { } ( z ) δ e (1 − e ) − ( z − α ) f { } ( z ) , which also follows by direct calculation from (13.1). Regarding ( β − z ) − + ( z − α ) − = δ ( β − z ) − ( z − α ) − , we can rewrite this, for all z ∈ C \ [ α, β ], as f ( z ) = s z − α · a /s β − z + f { } ( z ) a /s z − α − f { } ( z ) = − s α − z · δa /s ( β − z )( z − α ) + f { } ( z ) − a /s z − αa /s z − α − f { } ( z )= s h a /s z − α − f { } ( z ) i + ( α − z ) δa ( β − z )( z − α ) ( α − z ) h a /s z − α − f { } ( z ) i = s α − z + δa ( β − z )( z − α ) a /s z − α − f { } ( z ) , giving rise to a continued fraction expansion of functions f ∈ R ( C \ [ α, β ]).
14. Description via linear fractional transformation
We are now going to write the parametrization Ψ m from Theorem 13.6 as a matricial linearfractional transformation, as considered in Appendix E. The generating matrix-valued func-tion of this transformation is a composition of certain instances of the matrix polynomialsintroduced in Notations 9.6 and 10.6: Notation . Let ( s j ) κj =0 be a sequence of complex p × q matrices and let m ∈ Z ,κ . Thenlet ¨ V m := V V · · · V m , where V k := ¨ V a { k } ,s { k } for all k ∈ Z ,m − and V m := ¨ V s { m } .Regarding Notations 9.6 and 10.6, we see that ¨ V m is a complex ( p + q ) × ( p + q ) matrixpolynomial with deg ¨ V m ≤ m + 1). As a main result of the present paper, we now obtaina description of the set of all solutions to Problem FP [[ α, β ]; ( s j ) mj =0 ] via a matricial linearfractional transformation generated by that matrix polynomial: Theorem 14.2.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Denote by h ˜ w m ˜ x m ˜ y m ˜ z m i the q × q blockrepresentation of the restriction of ¨ V m onto C \ [ α, β ] :(a) Let Γ ∈ h ¨ P [ s { m } ] i and let [ G ; G ] ∈ Γ . Then det(˜ y m G + ˜ z m G ) does not vanishidentically in C \ [ α, β ] and the function F = ( ˜ w m G + ˜ x m G )(˜ y m G + ˜ z m G ) − (14.1) belongs to R q [[ α, β ]; ( s j ) mj =0 ] .(b) For each F ∈ R q [[ α, β ]; ( s j ) mj =0 ] , there exists a unique equivalence class Γ ∈ h ¨ P [ s { m } ] i such that (14.1) holds true for all [ G ; G ] ∈ Γ , namely the equivalence class h P ¨ G m ( F, ( s j ) mj =0 ) i of the m -th F α,β -transformed pair P ¨ G m ( F, ( s j ) mj =0 ) of F with re-spect to ( s j ) mj =0 . roof. Let the mappings ψ , ψ , . . . , ψ m be defined as in Theorem 13.6. Let V m := ¨ V s { m } and,in the case m ≥
1, let V k := ¨ V a { k } ,s { k } for all k ∈ Z ,m − . For all k ∈ Z ,m , let h ˜ a k ˜ b k ˜ c k ˜ d k i be the q × q block representation of the restriction of V k onto C \ [ α, β ]. Because of Proposition 3.58,we have ( s { k } j ) m − kj =0 ∈ F < q,m − k,α,β for all k ∈ Z ,m . Using Remarks 3.11 and 3.12, we thus easilyobtain a { k } ∈ C q × q H and R ( a { k } ) ⊆ R ( s { k } ) for all k ∈ Z ,m − and, furthermore, s { k } ∈ C q × q < forall k ∈ Z ,m . Consider now an arbitrary pair [ G ; G ] ∈ ¨ P [ s { m } ]. According to Notation 8.1,then [ G ; G ] belongs to PR q ( C \ [ α, β ]). In particular, G and G are C q × q -valued functionsmeromorphic in C \ [ α, β ]. Because of Proposition 7.9, the set A := P ( G ) ∪ P ( G ) ∪ ¨ E ([ G ; G ])is a discrete subset of C \ [ α, β ]. Hence, C \ ([ α, β ] ∪ A ) = ∅ . Let H m := ψ m ( h [ G ; G ] i ). Inview of Definition 9.5 and Notations 9.6 and 13.3, we conclude from Proposition 9.11 thendet[˜ c m ( z ) G ( z ) + ˜ d m ( z ) G ( z )] = 0 and h ˜ a m ( z ) G ( z ) + ˜ b m ( z ) G ( z ) ih ˜ c m ( z ) G ( z ) + ˜ d m ( z ) G ( z ) i − = [ ψ m ( h [ G ; G ] i )]( z ) = H m ( z )for all z ∈ C \ ([ α, β ] ∪ A ). In the case m ≥
1, let H k := ψ k ( H k +1 ) for all k ∈ Z ,m − . Byvirtue of Theorem 13.6, we have H m ∈ R q [[ α, β ]; ( s { m } j ) j =0 ] and, in the case m ≥
1, moreover H k ∈ R q [[ α, β ]; ( s { k } j ) m − kj =0 ] for all k ∈ Z ,m − .In the case m ≥
1, we now consider an arbitrary ℓ ∈ Z ,m − . Then H ℓ +1 ∈ R q ( C \ [ α, β ])and the R [ α, β ]-measure ¨ σ ℓ +1 of H ℓ +1 fulfills ¨ σ ℓ +1 ([ α, β ]) = s { ℓ +1 } . Using Proposition 4.15, weobtain in particular R ( H ℓ +1 ( z )) = R ( s { ℓ +1 } ) for all z ∈ C \ [ α, β ]. Regarding Definitions 3.55and 3.51, we infer furthermore R ( s { ℓ +1 } ) ⊆ R ( a { ℓ } ). Consequently, R ( H ℓ +1 ( z )) ⊆ R ( a { ℓ } )follows for all z ∈ C \ [ α, β ].Thus, in the case m ≥
1, Lemma 10.8 yields, in view of Notations 10.6 and 13.3, for all k ∈ Z ,m − , then det[˜ c k ( z ) H k +1 ( z ) + ˜ d k ( z )] = 0 and h ˜ a k ( z ) H k +1 ( z ) + ˜ b k ( z ) ih ˜ c k ( z ) H k +1 ( z ) + ˜ d k ( z ) i − = [ ψ k ( H k +1 )]( z ) = H k ( z )for all z ∈ C \ [ α, β ]. By virtue of ¨ V m = V V · · · V m , we can conclude from Proposition E.2hence det[˜ y m ( z ) G ( z ) + ˜ z m ( z ) G ( z )] = 0 and H ( z ) = [( ψ ◦ ψ ◦ · · · ◦ ψ m )( h [ G ; G ] i )]( z )= [ ˜ w m ( z ) G ( z ) + ˜ x m ( z ) G ( z )][˜ y m ( z ) G ( z ) + ˜ z m ( z ) G ( z )] − for all z ∈ C \ ([ α, β ] ∪A ). In particular, det(˜ y m G +˜ z m G ) does not vanish identically in C \ [ α, β ].Consequently, ( ˜ w m G + ˜ x m G )(˜ y m G + ˜ z m G ) − is a C q × q -valued function meromorphic in C \ [ α, β ]. By virtue of H ∈ R q [[ α, β ]; ( s j ) mj =0 ], the matrix-valued function H is holomorphicin C \ [ α, β ]. Since the set A is discrete, we can conclude from the identity theorem forholomorphic functions then( ψ ◦ ψ ◦ · · · ◦ ψ m )( h [ G ; G ] i ) = H = ( ˜ w m G + ˜ x m G )(˜ y m G + ˜ z m G ) − . The application of Theorem 13.6 completes the proof.
Remark . Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . In view of δ > δ m − d m = s { m } , and consequently R ( d m ) = R ( s { m } ) and N ( d m ) = N ( s { m } ). Accordingto Notation 8.1, hence ¨ P [ d m ] = ¨ P [ s { m } ]. 68ow we study separately three cases depending on the rank r of the matrix d m = δ − ( m − s { m } determining the amount of determinacy of the truncated moment problem in question. Wedistinguish between the case r = q , the case 1 ≤ r ≤ q −
1, and the case r = 0.First we consider the so-called non-degenerate case r = q . In view of Remark 3.66, this isexactly the case of ( s j ) mj =0 ∈ F ≻ q,m,α,β . For this situation Problem FP [[ α, β ]; ( s j ) mj =0 ] was alreadyconsidered in [10,11], applying Potapov’s method of fundamental matrix inequalities separatelyfor even and odd non-negative integers m , resp. The generating matrix-valued functions ofthe linear fractional transformation obtained there are matrix polynomials. For the scalar case q = 1 we refer to Krein/Nudelman [42, Ch. IV, § 7], where the generating matrix functionof the linear fractional transformation is built from orthogonal polynomials of first kind andsecond kind. Theorem 14.4.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β be such that the matrix d m is non-singular. Then all statements of Theorem 14.2 are valid with the class PR q ( C \ [ α, β ]) insteadof ¨ P [ s { m } ] .Proof. Use Remarks 14.3 and 8.2 and Theorem 14.2.Now we turn our attention to the degenerate, but not completely degenerate case 1 ≤ r ≤ q − Theorem 14.5.
Assume q ≥ . Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Let r := rank d m and assume ≤ r ≤ q − . Denote by h ˜ w m ˜ x m ˜ y m ˜ z m i the q × q block representation of the restrictionof ¨ V m onto C \ [ α, β ] . Let u , u , . . . , u q be an orthonormal basis of C q with { u , u , . . . , u r } ⊆R ( d m ) and let W := [ u , u , . . . , u q ] . Then:(a) Let [ g ; g ] ∈ PR r ( C \ [ α, β ]) . Then det[˜ y m W ( g ⊕ O ( q − r ) × ( q − r ) ) + ˜ z m W ( g ⊕ I q − r )] doesnot vanish identically in C \ [ α, β ] .(b) For each [ g ; g ] ∈ PR r ( C \ [ α, β ]) let S W, [ g ; g ] : C \ [ α, β ] → C q × q be defined by S W, [ g ; g ] := h ˜ w m W ( g ⊕ O ( q − r ) × ( q − r ) ) + ˜ x m W ( g ⊕ I q − r ) i × h ˜ y m W ( g ⊕ O ( q − r ) × ( q − r ) ) + ˜ z m W ( g ⊕ I q − r ) i − . Then Σ W : hPR r ( C \ [ α, β ]) i → R q [[ α, β ]; ( s j ) mj =0 ] defined by Σ W ( h [ g ; g ] i ) := S W, [ g ; g ] is well defined and bijective.Proof. Obviously, W is a unitary q × q matrix and U := [ u , u , . . . , u r ] is the left q × r blockof W . In view of dim R ( d m ) = r , we furthermore see that u , u , . . . , u r is an orthonormalbasis of R ( d m ). Due to Lemma 8.5, the mapping Γ U : hPR r ( C \ [ α, β ]) i → h ¨ P [ d m ] i defined byΓ U ( h [ f ; f ] i ) := h [ U f U ∗ ; U f U ∗ + P [ R ( d m )] ⊥ ] i is thus well defined and bijective. By virtue ofRemark 14.3, we have ¨ P [ d m ] = ¨ P [ s { m } ]. Consider now an arbitrary pair [ g ; g ] ∈ PR r ( C \ [ α, β ]). Then g and g are C r × r -valued functions, which are meromorphic in C \ [ α, β ]. Observethat ˜ w m , ˜ x m , ˜ y m , and ˜ z m , as restrictions of q × q matrix polynomials, are C q × q -valued functions,which are holomorphic in C \ [ α, β ]. Consequently, we can easily conclude that X U, [ g ; g ] := ˜ w m U g U ∗ + ˜ x m ( U g U ∗ + P [ R ( d m )] ⊥ )69nd Y U, [ g ; g ] := ˜ y m U g U ∗ + ˜ z m ( U g U ∗ + P [ R ( d m )] ⊥ )are C q × q -valued functions which are meromorphic in C \ [ α, β ] with P ( X U, [ g ; g ] ) ⊆ P ( g ) ∪P ( g )and P ( Y U, [ g ; g ] ) ⊆ P ( g ) ∪ P ( g ). Therefore, det Y U, [ g ; g ] is a complex-valued function whichis meromorphic in C \ [ α, β ] with P (det Y U, [ g ; g ] ) ⊆ P ( g ) ∪ P ( g ). Similarly, T W, [ g ; g ] := ˜ w m W ( g ⊕ O ( q − r ) × ( q − r ) ) + ˜ x m W ( g ⊕ I q − r )and R W, [ g ; g ] := ˜ y m W ( g ⊕ O ( q − r ) × ( q − r ) ) + ˜ z m W ( g ⊕ I q − r )are C q × q -valued functions which are meromorphic in C \ [ α, β ] with P ( T W, [ g ; g ] ) ⊆ P ( g ) ∪P ( g ) and P ( R W, [ g ; g ] ) ⊆ P ( g ) ∪ P ( g ), and det R W, [ g ; g ] is a complex-valued function whichis meromorphic in C \ [ α, β ] with P (det R W, [ g ; g ] ) ⊆ P ( g ) ∪ P ( g ). In view of [ g ; g ] ∈PR r ( C \ [ α, β ]), we have Γ U ( h [ g ; g ] i ) ∈ h ¨ P [ d m ] i = h ¨ P [ s { m } ] i . Due to Theorem 14.2(a),hence det Y U, [ g ; g ] does not vanish identically in C \ [ α, β ]. Regarding the identity theorem forholomorphic functions, then N := { ζ ∈ C \ ([ α, β ] ∪ P (det Y U, [ g ; g ] )) : det Y U, [ g ; g ] ( ζ ) = 0 } isa discrete subset of C \ [ α, β ]. Consequently, D := P ( g ) ∪ P ( g ) ∪ N is a discrete subset of C \ [ α, β ], which fulfills P ( X U, [ g ; g ] ) ∪ P ( Y U, [ g ; g ] ) ∪ P (det Y U, [ g ; g ] ) ⊆ D and P ( T W, [ g ; g ] ) ∪P ( R W, [ g ; g ] ) ∪ P (det R W, [ g ; g ] ) ⊆ D . In particular, C \ ([ α, β ] ∪ D ) is non-empty. Consider nowan arbitrary z ∈ C \ ([ α, β ] ∪ D ). Then det Y U, [ g ; g ] ( z ) = 0 and, furthermore, X U, [ g ; g ] ( z ) = ˜ w m ( z )[ U g ( z ) U ∗ ] + ˜ x m ( z ) h U g ( z ) U ∗ + P [ R ( d m )] ⊥ i ,Y U, [ g ; g ] ( z ) = ˜ y m ( z )[ U g ( z ) U ∗ ] + ˜ z m ( z ) h U g ( z ) U ∗ + P [ R ( d m )] ⊥ i ,T W, [ g ; g ] ( z ) = ˜ w m ( z ) W h g ( z ) ⊕ O ( q − r ) × ( q − r ) i + ˜ x m ( z ) W [ g ( z ) ⊕ I q − r ] , and R W, [ g ; g ] ( z ) = ˜ y m ( z ) W h g ( z ) ⊕ O ( q − r ) × ( q − r ) i + ˜ z m ( z ) W [ g ( z ) ⊕ I q − r ] . Consequently, using Lemma E.3(b), we obtain det R W, [ g ; g ] ( z ) = 0 and h X U, [ g ; g ] ( z ) ih Y U, [ g ; g ] ( z ) i − = h T W, [ g ; g ] ( z ) ih R W, [ g ; g ] ( z ) i − . In particular, det R W, [ g ; g ] does not vanish identically in C \ [ α, β ], showing (a). Therefore, S W, [ g ; g ] = T W, [ g ; g ] R − W, [ g ; g ] is a C q × q -valued function which is meromorphic in C \ [ α, β ] andholomorphic at z with S W, [ g ; g ] ( z ) = h X U, [ g ; g ] ( z ) ih Y U, [ g ; g ] ( z ) i − = (cid:16) ˜ w m ( z )[ U g ( z ) U ∗ ] + ˜ x m ( z ) h U g ( z ) U ∗ + P [ R ( d m )] ⊥ i(cid:17) × (cid:16) ˜ y m ( z )[ U g ( z ) U ∗ ] + ˜ z m ( z ) h U g ( z ) U ∗ + P [ R ( d m )] ⊥ i(cid:17) − . (14.2)Since (14.2) holds true for all z ∈ C \ ([ α, β ] ∪ D ) and the set D is discrete, we can concludefrom the identity theorem for holomorphic functions that S W, [ g ; g ] = h ˜ w m ( U g U ∗ ) + ˜ x m ( U g U ∗ + P [ R ( d m )] ⊥ ) i × h ˜ y m ( U g U ∗ ) + ˜ z m ( U g U ∗ + P [ R ( d m )] ⊥ ) i − . P ; Q ] ∈ ¨ P [ d m ] = ¨ P [ s { m } ], we can consider thefunction F [ P ; Q ] := ( ˜ w m P +˜ x m Q )(˜ y m P +˜ z m Q ) − . In view of Definition 7.11, it is readily checkedthat, given two arbitrary pairs [ P ; Q ] , [ P ; Q ] ∈ ¨ P [ d m ], the equivalence [ P ; Q ] ∼ [ P ; Q ]implies F [ P ; Q ] = F [ P ; Q ] . According to Theorem 14.2, thus the mapping Π : h ¨ P [ d m ] i →R q [[ α, β ]; ( s j ) mj =0 ] defined by Π( h [ P ; Q ] i ) := F [ P ; Q ] is well defined and bijective. By virtue ofthat we have already shown, we getΣ W ( h [ g ; g ] i ) = S W, [ g ; g ] = F [ Ug U ∗ ; Ug U ∗ + P [ R ( d m )] ⊥ ] = Π (cid:16) h [ U g U ∗ ; U g U ∗ + P [ R ( d m )] ⊥ ] i (cid:17) = Π(Γ U ( h [ g ; g ] i )) . Since this holds true for all pairs [ g ; g ] ∈ PR r ( C \ [ α, β ]), we thus verified that Σ W = Π ◦ Γ U is well defined and bijective, i. e., (b) holds true.Now we treat the completely degenerate case r = 0. We will see in particular that in thissituation the solution is unique. Theorem 14.6.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ] -interval lengths ( d j ) mj =0 . Then R q [[ α, β ]; ( s j ) mj =0 ] consists of exactly one element if and only if d m = O q × q . Inthis case, R q [[ α, β ]; ( s j ) mj =0 ] = { ˜ x m ˜ z − m } , where h ˜ w m ˜ x m ˜ y m ˜ z m i denotes the q × q block representationof the restriction of ¨ V m onto C \ [ α, β ] .Proof. Assume d m = O q × q . For each ℓ ∈ { , } let ( s ℓ,j ) m +1 j =0 be given by s ℓ,j := ( s j , if 0 ≤ j ≤ m a m + ℓ d m , if j = m + 1 . Due to Corollary 3.35, then { ( s ,j ) m +1 j =0 , ( s ,j ) m +1 j =0 } ⊆ F < q,m +1 ,α,β . By virtue of Proposition 5.6,thus R q [[ α, β ]; ( s ℓ,j ) m +1 j =0 ] is non-empty for each ℓ ∈ { , } . Consequently, we can choose, foreach ℓ ∈ { , } , a function F ℓ ∈ R q [[ α, β ]; ( s ℓ,j ) m +1 j =0 ]. Observe that, for each ℓ ∈ { , } , the R [ α, β ]-measure σ ℓ of F ℓ belongs to M < q,m +1 [[ α, β ]; ( s ℓ,j ) m +1 j =0 , =], implying R [ α,β ] x m +1 σ ℓ (d x ) = s ℓ,m +1 . Because of s ,m +1 − s ,m +1 = d m = O q × q , we have σ = σ and hence F = F . Sincethe functions F and F both belong to R q [[ α, β ]; ( s j ) mj =0 ], therefore this set consists of at leasttwo elements. Thus, we can reversely conclude that if the set R q [[ α, β ]; ( s j ) mj =0 ] consists ofexactly one element, then necessarily d m = O q × q follows.Assume d m = O q × q . By virtue of Lemma 8.5, then h ¨ P [ d m ] i consists of exactly one element,namely the equivalence class h [ G ; G ] i of the pair [ G ; G ] built from the functions G , G : C \ [ α, β ] → C q × q defined by G ( z ) := O q × q and G ( z ) := I q . Due to Remark 14.3, we have¨ P [ d m ] = ¨ P [ s { m } ]. Consequently, D ¨ P [ s { m } ] E = D ¨ P [ d m ] E = {h [ G ; G ] i} (14.3)follows. Because of Proposition 5.6, the set R q [[ α, β ]; ( s j ) mj =0 ] is non-empty. Consider anarbitrary F ∈ R q [[ α, β ]; ( s j ) mj =0 ]. Taking into account (14.3), we can infer from Theorem 14.2then F = ( ˜ w m G + ˜ x m G )(˜ y m G + ˜ z m G ) − = ˜ x m ˜ z − m , implying R q [[ α, β ]; ( s j ) mj =0 ] = { ˜ x m ˜ z − m } . In particular, R q [[ α, β ]; ( s j ) mj =0 ] consists of exactly oneelement. 71n view of Propositions 5.5 and 3.28 and Remark 3.25, we obtain from Theorem 14.6 imme-diately the following two results: Corollary 14.7.
Let n ∈ N and let ( s j ) n +1 j =0 ∈ F < q, n +1 ,α,β . Then M < q, n +1 [[ α, β ]; ( s j ) n +1 j =0 , =] consists of exactly one element if and only if R ( L α,n, • ) ∩ R ( L • ,n,β ) = { O q × } . Corollary 14.8.
Let n ∈ N and let ( s j ) nj =0 ∈ F < q, n,α,β . Then M < q, n [[ α, β ]; ( s j ) nj =0 , =] consistsof exactly one element if and only if R ( L n ) ∩ R ( L α,n − ,β ) = { O q × } . We end this section with a necessary condition for unique solvability of Prob-lem MP [[ α, β ]; ( s j ) mj =0 , =]: Corollary 14.9.
Let m ∈ N and let ( s j ) mj =0 be a sequence of complex q × q matrices suchthat M < q,m [[ α, β ]; ( s j ) mj =0 , =] consists of exactly one element. Then rank A m + rank B m ≤ q .Proof. Due to Theorem 3.5, we have ( s j ) mj =0 ∈ F < q,m,α,β . In view of Proposition 5.5, we obtainfrom Theorem 14.6 thus d m = O q × q , i. e. rank d m = 0. Since rank d m − ≤ q holds true, we caninfer by virtue of Corollary 3.30 then rank A m + rank B m ≤ q .We are now going to factorize ¨ V m in a way alternative to Notation 14.1. In a first step, wederive by virtue of Lemma 10.17 a connection between ¨ V m and ¨ V m − , which against the back-ground of Theorem 14.2 correlates the solution sets: R q [[ α, β ]; ( s j ) mj =0 ] ⊆ R q [[ α, β ]; ( s j ) m − j =0 ]To that end, we make use of the sequences ( a { k } j ) κ − − kj =0 and ( b { k } j ) κ − − kj =0 given by a { k } j := − αs { k } j + s { k } j +1 and b { k } j := βs { k } j − s { k } j +1 , resp., i. e., the sequences built from the k -th F α,β -transform ( s { k } j ) κ − kj =0 according to Notation 3.2: Lemma 14.10.
Suppose κ ≥ . Let ( s j ) κj =0 ∈ F < q,κ,α,β . Then ¨ V m = ¨ V m − ¨ U a { m − } ,s { m − } forall m ∈ Z ,κ .Proof. Consider an arbitrary m ∈ Z ,κ . In view of Notation 14.1 it is sufficient to verify theidentity ¨ V A,M ¨ V D = ¨ V M ¨ U A,M with A := a { m − } , M := s { m − } , and D := s { m } .Due to Proposition 3.58, the sequence ( s { m − } j ) κ − m +1 j =0 belongs to F < q,κ − m +1 ,α,β . Using Re-marks 3.11 and 3.12, we can hence infer A, M ∈ C q × q H and R ( A ) ⊆ R ( M ). Let B := δM − A .Because of Remark 3.3, then B = δs { m − } − a { m − } = b { m − } . According to Definitions 3.21and 3.51, denote by ( d { m − } j ) κ − m +1 j =0 the sequence of [ α, β ]-interval lengths associated with( s { m − } j ) κ − m +1 j =0 and by ( t j ) κ − mj =0 the F α,β -transform of ( s { m − } j ) κ − m +1 j =0 , resp. Remark 3.10 yields( s { m − } j ) κ − m +1 j =0 ∈ D q × q,κ − m +1 . By virtue of Remark 3.23 and Lemma 3.53, we can henceinfer AM † B = d { m − } = t . In view of Definition 3.55, we have t = s { m } = D . Conse-quently, D = AM † B follows. Thus, we can apply Lemma 10.17 to obtain [ ¨ V A,M ( z )][ ¨ V D ( z )] =[ ¨ V M ( z )][ ¨ U A,M ( z )] for all z ∈ C .The above mentioned factorization alternative to Notation 14.1 results from Lemma 14.10now by means of mathematical induction: Proposition 14.11.
Let ( s j ) κj =0 ∈ F < q,κ,α,β and let m ∈ Z ,κ . Then ¨ V m = U U · · · U m ,where U ℓ := ¨ U a { ℓ − } ,s { ℓ − } for all ℓ ∈ Z ,m and U := ¨ V s . emma 14.12. Suppose κ ≥ . Let ( s j ) κj =0 ∈ F < q,κ,α,β with F α,β -parameter sequence ( f j ) κj =0 and sequence of [ α, β ] -interval lengths ( d j ) κj =0 . Let k ∈ Z ,κ − and let z ∈ C . Then ¨ U a { k } ,s { k } ( z ) = (cid:2) U U U U (cid:3) , where U = d k h ( β − z ) f † k +1 f k +1 d † k f k +2 + ( z − α )( I q − f † k +1 f k +1 ) d † k f k +1 i d † k +1 , U = δ k f k +2 ,U = − ( β − z )( z − α ) δ − k +1 d † k f k +1 d † k +1 , U = ( β − z ) δ h ( I q − d † k d k ) + d † k f k +1 i . Proof.
Let A := a { k } , let M := s { k } , and let B := δM − A . Because of Remark 3.3, then B = δs { k } − a { k } = b { k } . Corollary 3.60 shows M = δ k − d k . Furthermore, Proposition 3.59yields A = a { k } = δ k f k +1 and B = δ k f k +2 . Let D := AM † B and denote by ( d { k } j ) κ − kj =0 the sequence of [ α, β ]-interval lengths associated with ( s { k } j ) κ − kj =0 . Due to Proposition 3.58 wehave ( s { k } j ) κ − kj =0 ∈ F < q,κ − k,α,β . Because of Remark 3.10, in particular ( s { k } j ) κ − kj =0 ∈ D q × q,κ − k .Consequently, from Remark 3.23 we conclude d { k } = D . Proposition 3.61, then implies D = δ k d k +1 . Using Remarks A.15 and A.13 and taking into account δ >
0, Notation 10.15, and(9.2), the assertion follows.
15. On the sets R q [[ α, β ]; ( s j ) m +1 j =0 ] in the case of [ α, β ] -completelydegenerate extensions of a sequence ( s j ) mj =0 ∈ F < q,m,α,β In this section, we study [ α, β ]-completely degenerate extensions of a sequence ( s j ) mj =0 ∈F < q,m,α,β . First we recall the notion of [ α, β ]-completely degenerate sequences belonging to F < q,m,α,β and a characterization of this class of sequences. Definition 15.1 ( [27, Def. 10.24]) . Let ℓ ∈ N and let ( s j ) ℓj =0 ∈ F < q,ℓ,α,β with sequence of[ α, β ]-interval lengths ( d j ) ℓj =0 given in Definition 3.21. Then ( s j ) ℓj =0 is called [ α, β ] -completelydegenerate if d ℓ = O q × q . We denote by F < , cd q,ℓ,α,β the set of all sequences ( s j ) ℓj =0 ∈ F < q,ℓ,α,β whichare [ α, β ]-completely degenerate. Proposition 15.2 (cf. [28, Prop. 6.38]) . Let ℓ ∈ N and let ( s j ) ℓj =0 ∈ F < q,ℓ,α,β with [ α, β ] -intervalparameter sequence ( e j ) ℓj =0 given in Definition 3.42. Then ( s j ) ℓj =0 is [ α, β ] -completely degen-erate if and only if e ℓ = e ℓ . Observe that in the situation of Proposition 15.2, due to e ℓ < O q × q , we have e ∗ ℓ = e ℓ andthus the condition e ℓ = e ℓ is equivalent to e ℓ being a transformation matrix corresponding toan orthogonal projection, i. e., e ℓ = P R ( e ℓ ) .Against the background of Proposition 15.2, we are looking now for a description of the set { s m +1 ∈ C q × q : ( s j ) m +1 j =0 ∈ F < , cd q,m +1 ,α,β } . We will show that this set stands in a bijective correspondence to the set of all linear subspacesof R ( d m ). Notation . Let m ∈ N , let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ]-interval lengths( d j ) mj =0 , and let U be a linear subspace of R ( d m ). Then let s m, U := b m − d / m P U d / m if m iseven, and s m, U := a m + d / m P U d / m if m is odd.73 xample 15.4. Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ]-interval lengths( d j ) mj =0 . Let U := { O q × } and let U := R ( d m ). Then s m, U = b m and s m, U = a m if m is even,and s m, U = a m and s m, U = b m if m is odd.Indeed, we have P U = O q × q and, in view of R ( d / m ) = R ( d m ), furthermore P U = P R ( d / m ) .Consequently, d / m P U d / m = O q × q and d / m P U d / m = d m . Since, according to Definition 3.21,we have d m = b m − a m , the assertions follow by virtue of Notation 15.3. Proposition 15.5.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Then:(a) Let U be a linear subspace of R ( d m ) and let s m +1 := s m, U . Then ( s j ) m +1 j =0 ∈ F < , cd q,m +1 ,α,β .Furthermore, f m +1 = d / m ( I q − P U ) d / m , f m +2 = d / m P U d / m , and e m +1 = P U .(b) Let s m +1 ∈ C q × q be such that ( s j ) m +1 j =0 ∈ F < , cd q,m +1 ,α,β . Then there exists a linear subspace U of R ( d m ) such that s m +1 = s m, U , namely U = R ( e m +1 ) .(c) Let U and V be linear subspaces of R ( d m ) . Then U = V if and only if s m, U = s m, V .Proof. First observe that d m belongs to C q × q < , due to Proposition 3.27. Let D := d / m .(a) By virtue of Definitions 3.36 and 3.24 and Notation 15.3, we have f m +2 = f n +2 = B n +1 = B m +1 = b m − s m +1 = D P U D in the case m = 2 n for some n ∈ N , and f m +2 = f n +4 = A n +2 = A m +1 = s m +1 − a m = D P U D in the case m = 2 n + 1 for some n ∈ N . Using Remark 3.39, we can infer then f m +1 = d m − f m +2 = DD − D P U D = D ( I q − P U ) D . Because of O q × q P U I q andRemark A.25, consequently the matrices f m +2 and f m +1 are both non-negative Hermitian.From Proposition 3.38, we can conclude now ( s j ) m +1 j =0 ∈ F < q,m +1 ,α,β . According to Defini-tion 3.42, then e m +1 = D † f m +2 D † = D † D P U DD † . Because of R ( d m ) = R ( D ), we have U ⊆ R ( D ). Remarks A.11 and A.20 then yield DD † P U = P U . Furthermore D ∗ = D im-plies D † D = DD † , by virtue of Remark A.18. Hence, D † D P U = P U follows. Taking account P ∗U = P U and (3.3), we furthermore obtain P U = ( DD † P U ) ∗ = P U DD † . Consequently, weconclude e m +1 = D † D P U DD † = P U . In view of P U = P U , the application of Proposition 15.2provides then ( s j ) m +1 j =0 ∈ F < , cd q,m +1 ,α,β .(b) According to Proposition 15.2, we have e m +1 = e m +1 . From Proposition 3.46 and (3.6),we get e ∗ m +1 = e m +1 and O q × q e m +1 P R ( d m ) . Consequently, e m +1 = P U with U := R ( e m +1 ).Furthermore, P U P R ( d m ) , implying U ⊆ R ( d m ). Lemma 3.43 yields f m +2 = D e m +1 D . Byvirtue of Definitions 3.36 and 3.24, we have then D P U D = f m +2 = f n +2 = B n +1 = B m +1 = b m − s m +1 in the case m = 2 n for some n ∈ N , and D P U D = f m +2 = f n +4 = A n +2 = A m +1 = s m +1 − a m in the case m = 2 n + 1 for some n ∈ N . In view of Notation 15.3, hence s m +1 = s m, U follows.(c) Obviously, U = V implies s m, U = s m, V , according to Notation 15.3. Conversely, suppose s m, U = s m, V . From Notation 15.3, then D P U D = D P V D follows. By the same reasoning as inthe proof of part (a), we can infer D † D P U DD † = P U and D † D P V DD † = P V . Consequently, P U = P V , implying U = V . 74 otation . Let m ∈ N , let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ]-interval lengths( d j ) mj =0 , and let U be a linear subspace of R ( d m ). Then let X m, U , Y m, U : C \ [ α, β ] → C q × q bedefined by X m, U ( z ) := δ m − d / m P U d / m , Y m, U ( z ) := ( β − z )[ I q − ( d / m ) † P U d / m ] . Example 15.7.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ]-interval lengths( d j ) mj =0 . Let U := { O q × } and let U := R ( d m ). For all z ∈ C \ [ α, β ], then X m, U ( z ) = O q × q and Y m, U ( z ) := ( β − z ) I q as well as X m, U ( z ) := δ m − d m and Y m, U ( z ) := ( β − z ) P N ( d m ) . Indeed,as in the proof of Example 15.4, we obtain d / m P U d / m = O q × q and d / m P U d / m = d m . Takinginto account d † m d / m = ( d / m ) † , then ( d / m ) † P U d / m = O q × q and ( d / m ) † P U d / m = d † m d m follow.Hence, I q − ( d / m ) † P U d / m = I q and, in view of Remark A.18, furthermore I q − ( d / m ) † P U d / m = P N ( d m ) . Now, the assertions follow by virtue of Notation 15.6. Remark . Let m ∈ N , let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ]-interval lengths ( d j ) mj =0 ,and let U be a linear subspace of R ( d m ). In view of Notations 15.6 and 8.1 and R ( d m ) = R ( d / m )as well as Example 8.7, then [ X m, U ; Y m, U ] ∈ ¨ P [ d m ]. Lemma 15.9.
Let m ∈ N , let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ] -interval lengths ( d j ) mj =0 , and let U be a linear subspace of R ( d m ) . Denote by h ˜ w m ˜ x m ˜ y m ˜ z m i the q × q block represen-tation of the restriction of ¨ V m onto C \ [ α, β ] . Then the function det(˜ y m X m, U + ˜ z m Y m, U ) doesnot vanish identically.Proof. In view of Remarks 15.8 and 14.3, this is a consequence of Theorem 14.2(a).Lemma 15.9 shows that the following notation is correct.
Notation . Let m ∈ N , let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ]-interval lengths( d j ) mj =0 , and let U be a linear subspace of R ( d m ) Denote by h ˜ w m ˜ x m ˜ y m ˜ z m i the q × q block represen-tation of the restriction of ¨ V m onto C \ [ α, β ]. Then let S m, U := ( ˜ w m X m, U + ˜ x m Y m, U )(˜ y m X m, U + ˜ z m Y m, U ) − . Lemma 15.11.
Let m ∈ N , let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ] -interval lengths ( d j ) mj =0 , and let U be a linear subspace of R ( d m ) . Then S m, U ∈ R q [[ α, β ]; ( s j ) mj =0 ] .Proof. In view of Remarks 15.8 and 14.3 and Notation 15.10, this is a consequence of Theo-rem 14.2(a).Given a sequence ( s j ) m +1 j =0 ∈ F < q,m +1 ,α,β , now we look for the q × q matrix polynomials in thefour q × q blocks of the 2 q × q matrix polynomial ¨ V m +1 defined in Notation 14.1. Lemma 15.12.
Let m ∈ N and let ( s j ) m +1 j =0 ∈ F < q,m +1 ,α,β with F α,β -parameter sequence ( f j ) m +2 j =0 and sequence of [ α, β ] -interval lengths ( d j ) m +1 j =0 . Denote by (cid:2) w m x m y m z m (cid:3) and h w m +1 x m +1 y m +1 z m +1 i he q × q block representations of ¨ V m and ¨ V m +1 , resp. For all z ∈ C , then w m +1 ( z ) = (cid:26) w m ( z ) d m h ( β − z ) f † m +1 f m +1 d † m f m +2 + ( z − α )( I q − f † m +1 f m +1 ) d † m f m +1 i − ( β − z )( z − α ) δ − m +1 x m ( z ) d † m f m +1 (cid:27) d † m +1 , x m +1 ( z ) = δ (cid:16) δ m − w m ( z ) f m +2 + ( β − z ) x m ( z ) h ( I q − d † m d m ) + d † m f m +1 i(cid:17) , y m +1 ( z ) = (cid:26) y m ( z ) d m h ( β − z ) f † m +1 f m +1 d † m f m +2 + ( z − α )( I q − f † m +1 f m +1 ) d † m f m +1 i − ( β − z )( z − α ) δ − m +1 z m ( z ) d † m f m +1 (cid:27) d † m +1 , and z m +1 ( z ) = δ (cid:16) δ m − y m ( z ) f m +2 + ( β − z ) z m ( z ) h ( I q − d † m d m ) + d † m f m +1 i(cid:17) . Proof.
From Lemma 14.10 we obtain ¨ V m +1 = ¨ V m ¨ U a { m } ,s { m } . Using the q × q block represen-tations of ¨ V m +1 and ¨ V m as well as Lemma 14.12, a straightforward calculation completes theproof.Now we specify Lemma 15.12 for the case of a sequence ( s j ) m +1 j =0 ∈ F < , cd q,m +1 ,α,β . Lemma 15.13.
Let m ∈ N and let ( s j ) m +1 j =0 ∈ F < , cd q,m +1 ,α,β with F α,β -parameter sequence ( f j ) m +2 j =0 and sequence of [ α, β ] -interval lengths ( d j ) m +1 j =0 . Denote by (cid:2) w m x m y m z m (cid:3) and h w m +1 x m +1 y m +1 z m +1 i the q × q block representations of ¨ V m and ¨ V m +1 , resp. For all z ∈ C , then w m +1 ( z ) = O q × q and y m +1 ( z ) = O q × q as well as x m +1 ( z ) = δ (cid:16) δ m − w m ( z ) f m +2 + ( β − z ) x m ( z ) h ( I q − d † m d m ) + d † m f m +1 i(cid:17) and z m +1 ( z ) = δ (cid:16) δ m − y m ( z ) f m +2 + ( β − z ) z m ( z ) h ( I q − d † m d m ) + d † m f m +1 i(cid:17) . Proof.
According to Definition 15.1, we have ( s j ) m +1 j =0 ∈ F < q,m +1 ,α,β and d m +1 = O q × q . Theapplication of Lemma 15.12 completes the proof.Let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ]-interval lengths ( d j ) mj =0 and let U be a linearsubspace of R ( d m ). Using Notation 15.3 to define s m +1 := s m, U , we determine now the set R q [[ α, β ]; ( s j ) m +1 j =0 ]. Proposition 15.14.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ] -intervallengths ( d j ) mj =0 . Let U be a linear subspace of R ( d m ) and let s m +1 := s m, U . Then R q [[ α, β ]; ( s j ) m +1 j =0 ] = { S m, U } , where S m, U is given via Notation 15.10.Proof. Denote by ( f j ) m +2 j =0 the F α,β -parameter sequence of ( s j ) m +1 j =0 . Because of Proposi-tion 15.5(a), we have ( s j ) m +1 j =0 ∈ F < , cd q,m +1 ,α,β and furthermore f m +1 = d / m ( I q − P U ) d / m and f m +2 = d / m P U d / m . In particular, d † m f m +1 = d † m d m − d † m d / m P U d / m and, in view of76 † m d / m = ( d / m ) † , consequently ( I q − d † m d m ) + d † m f m +1 = I q − ( d / m ) † P U d / m . By virtue of Nota-tion 15.6, for all z ∈ C \ [ α, β ], hence δ m − f m +2 = X m, U ( z ) and ( β − z )[( I q − d † m d m )+ d † m f m +1 ] = Y m, U ( z ). Denote by h ˜ w m ˜ x m ˜ y m ˜ z m i and h ˜ w m +1 ˜ x m +1 ˜ y m +1 ˜ z m +1 i the q × q block representations of the re-strictions of ¨ V m and ¨ V m +1 , resp., onto C \ [ α, β ]. From Lemma 15.13 we can infer then˜ x m +1 ( z ) = δ [ ˜ w m ( z ) X m, U ( z ) + ˜ x m ( z ) Y m, U ( z )] and ˜ z m +1 ( z ) = δ [˜ y m ( z ) X m, U ( z ) + ˜ z m ( z ) Y m, U ( z )]for all z ∈ C \ [ α, β ]. In view of δ >
0, Lemma 15.9, and Notation 15.10, we see that det ˜ z m +1 does not identically vanish and that S m, U = ˜ x m +1 ˜ z − m +1 . By virtue of Definition 15.1, we canapply Theorem 14.6 to complete the proof.Now we recall some facts and notions from [27]. Proposition 15.15 ( [27, Prop. 11.4]) . Let m ∈ N , let ( s j ) mj =0 ∈ F < q,m,α,β , and let s m +1 ∈{ a m , b m } . Then ( s j ) m +1 j =0 ∈ F < , cd q,m +1 ,α,β . Definition 15.16 (cf. [27, Def. 11.5]) . Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Let the sequence( s j ) ∞ j = m +1 be recursively defined by s j := a j − (resp. s j := b j − ). Then ( s j ) ∞ j =0 is called the lower (resp. upper ) [ α, β ]-completely degenerate sequence associated with ( s j ) mj =0 . Proposition 15.17 (cf. [27, Prop. 211]) . Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Denoteby ( s j ) ∞ j =0 and ( s j ) ∞ j =0 the lower and upper [ α, β ] -completely degenerate sequence associatedwith ( s j ) mj =0 , resp. Then the set M < q, ∞ [[ α, β ]; ( s j ) ∞ j =0 , =] contains exactly one element σ m andthe set M < q, ∞ [[ α, β ]; ( s j ) ∞ j =0 , =] contains exactly one element σ m . Definition 15.18 ( [27, Def. 12.4]) . Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Then the non-negative Hermitian q × q measure σ m (resp. σ m ) is called the lower (resp. upper ) CD-measureassociated with ( s j ) mj =0 and [ α, β ].Now we are interested in the [ α, β ]-Stieltjes transforms of σ m and σ m , resp. Definition 15.19.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Denote by σ m (resp. σ m ) the lower(resp. upper) CD-measure associated with ( s j ) mj =0 and [ α, β ]. Let S m be the [ α, β ]-Stieltjestransform of σ m and let S m be the [ α, β ]-Stieltjes transform of σ m . Then we call S m (resp. S m ) the lower (resp. upper ) R [ α, β ] -function associated with ( s j ) mj =0 .In Theorem 14.2, we obtained a complete description of the set R q [[ α, β ]; ( s j ) mj =0 ] of[ α, β ]-Stieltjes transforms of measures belonging to M < q,m [[ α, β ]; ( s j ) mj =0 , =]. Now we are inter-ested in the position of S m and S m in the set R q [[ α, β ]; ( s j ) mj =0 ]. In particular, we determinethe pairs [ X m ; Y m ] ∈ ¨ P [ s { m } ] and [ X m ; Y m ] ∈ ¨ P [ s { m } ] which correspond to S m and S m , resp.,according to Theorem 14.2(b). It can be expected that these pairs possess certain extremalproperties within the set ¨ P [ s { m } ].The preceding considerations lead us now quickly to explicit expressions for S m and S m . Proposition 15.20.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ] -intervallengths ( d j ) mj =0 . Let U := { O q × } and let U := R ( d m ) . Then S m = S m, U and S m = S m, U if m is even, and S m = S m, U and S m = S m, U if m is odd.Proof. Denote by ( s j ) ∞ j =0 and ( s j ) ∞ j =0 the lower and upper [ α, β ]-completely degenerate se-quence associated with ( s j ) mj =0 , resp. From Proposition 15.17 we can conclude σ m ∈ < q,m +1 [[ α, β ]; ( s j ) m +1 j =0 , =] and σ m ∈ M < q,m +1 [[ α, β ]; ( s j ) m +1 j =0 , =]. Remark 5.7 then shows S m ∈ R q [[ α, β ]; ( s j ) m +1 j =0 ] and S m ∈ R q [[ α, β ]; ( s j ) m +1 j =0 ].First consider the case m = 2 n with some n ∈ N . Because of Example 15.4, we have then s m, U = b m and s m, U = a m . In view of Definition 15.16, we can thus apply Proposition 15.14to the sequences ( s j ) m +1 j =0 and ( s j ) m +1 j =0 , resp., to obtain R q [[ α, β ]; ( s j ) m +1 j =0 ] = { S m, U } and R q [[ α, β ]; ( s j ) m +1 j =0 ] = { S m, U } . Thus, S m = S m, U and S m = S m, U follow.Now consider the case m = 2 n +1 with some n ∈ N . Because of Example 15.4, we have then s m, U = a m and s m, U = b m . In view of Definition 15.16, we can thus apply Proposition 15.14to the sequences ( s j ) m +1 j =0 and ( s j ) m +1 j =0 , resp., to obtain R q [[ α, β ]; ( s j ) m +1 j =0 ] = { S m, U } and R q [[ α, β ]; ( s j ) m +1 j =0 ] = { S m, U } . Thus, S m = S m, U and S m = S m, U follow.Finally, we want to indicate the announced extremal properties of the pairs [ X m ; Y m ] and[ X m ; Y m ] from ¨ P [ d m ] which correspond to S m and S m according to Theorem 14.2(b). Remark . If we look back to Proposition 15.14 and Example 15.7 and consider the corre-sponding pairs [ X m ; Y m ] and [ X m ; Y m ] belonging to ¨ P [ d m ], then it should be mentioned thatthese pairs consist of C q × q -valued functions in C \ [ α, β ], which have extremal rank properties.Indeed, the function X m satisfies rank X m = rank d m , which is the maximal possible rank of a q × q matrix-valued function X with R ( X ( z )) ⊆ R ( d m ) for all points z ∈ C \ [ α, β ] which arepoints of holomorphy of X , whereas the function X m has rank 0 which is clearly the minimalpossible rank.
16. On the [ α, β ] -Stieltjes transform of the central solutioncorresponding to a sequence ( s j ) mj =0 ∈ F < q,m,α,β At the beginning of this section we state the necessary background information.Recall that the sequences ( a j ) κj =0 and ( b j ) κj =0 were introduced in Definition 3.18. Definition 16.1 (cf. [27, Def. 10.11]) . If ( s j ) κj =0 is a sequence of complex p × q matrices, thenwe call ( m j ) κj =0 given by m j := ( a j + b j ) the sequence of [ α, β ] -interval mid points associatedwith ( s j ) κj =0 . Definition 16.2 (cf. [27, Def. 10.33]) . Let ( s j ) κj =0 be a sequence of complex p × q matriceswith sequence of [ α, β ]-interval mid points ( m j ) κj =0 . Assume κ ≥ k ∈ Z ,κ . Then( s j ) κj =0 is said to be [ α, β ] -central of order k if s j = m j − for all j ∈ Z k,κ . Definition 16.3 ( [27, Def. 11.9]) . Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Let the sequence( s j ) ∞ j = m +1 be recursively defined by s j := m j − , where m j − is given by Definition 16.1. Then( s j ) ∞ j =0 is called the [ α, β ] -central sequence associated with ( s j ) mj =0 . Proposition 16.4 ( [27, Prop. 11.10]) . Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Then the [ α, β ] -central sequence associated with ( s j ) mj =0 is [ α, β ] -non-negative definite and [ α, β ] -centralof order m + 1 . Proposition 16.5.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Denote by (˚ s j ) ∞ j =0 the [ α, β ] -centralsequence associated with ( s j ) mj =0 . Then the set M < q, ∞ [[ α, β ]; (˚ s j ) ∞ j =0 , =] contains exactly oneelement ˚ σ m .Proof. Combine Propositions 16.4 and 3.6. 78roposition 16.5 leads us to the following notion.
Definition 16.6.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Then the non-negative Hermitian q × q measure ˚ σ m mentioned in Proposition 16.5 is called the [ α, β ] -central measure associatedwith ( s j ) mj =0 . Definition 16.7.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Denote by ˚ σ m the [ α, β ]-centralmeasure associated with ( s j ) mj =0 . Then the [ α, β ]-Stieltjes transform ˚ S m of ˚ σ m is call the[ α, β ] -central function associated with ( s j ) mj =0 .Our next goal is now to determine the position of the [ α, β ]-Stieltjes transform of ˚ σ m withinthe parametrization of R q [[ α, β ]; ( s j ) mj =0 ] obtained in Theorem 14.2. In order to realize this plan,we continue our investigations in [29, Sec. 10] where we studied a Schur type transformation formatrix measures on [ α, β ] which transforms the concrete matrix measure under considerationin accordance with the Schur type algorithm considered in Definition 3.55, which has to beapplied to the corresponding moment sequence. Definition 16.8 ( [29, Def. 10.6]) . Let σ ∈ M < q ([ α, β ]) with sequence of power moments( s ( σ ) j ) ∞ j =0 and let k ∈ N . Then σ is called central of order k if ( s ( σ ) j ) ∞ j =0 is [ α, β ]-central of order k . Against the background of centrality of measures on [ α, β ], we consider now the scalar case,in particular the following object discussed in [29, Sec. 10]. Notation . Let a, b ∈ R with a < b and let ν [ a,b ] : B [ a,b ] → [0 , ∞ ) be the arcsine distribution on [ a, b ] given by ν [ a,b ] ( B ) := R B h d λ , where λ : B [ a,b ] → [0 , ∞ ) is the Lebesgue measure on [ a, b ]and h : [ a, b ] → [0 , ∞ ) is defined by h ( x ) := 0 if x ∈ { a, b } and by h ( x ) := [ π p ( x − a )( b − x )] − if x ∈ ( a, b ).Now we turn our attention to the ordinary and canonical moments of ν [ a,b ] . Example 16.10.
Let a, b ∈ R with a < b . Then ν [ a,b ] ∈ M < ([ a, b ]). Denote by ( s j ) ∞ j =0 the se-quence of power moments associated with ν [ a,b ] and by ( e j ) ∞ j =0 the sequence of (matricial) canon-ical moments associated with ν [ a,b ] via Definition 6.1. Then s j = P jk =0 (cid:0) jk (cid:1)(cid:0) kk (cid:1) − k ( b − a ) k a j − k for all j ∈ N . In particular, ν [ a,b ] ([ a, b ]) = 1 and R [ a,b ] tν [ a,b ] (d t ) = ( a + b ) /
2. Furthermore, e = 1 and e j = 1 / j ∈ N .Indeed, the measure µ := ν [0 , is a probability measure on [0 ,
1] with moments R [0 , x k µ (d x ) = (cid:0) kk (cid:1) − k for all k ∈ N (see, e. g. [36, formula (25.1)]) and (classical) canonicalmoments p k = 1 / k ∈ N (see, e. g. [12, Example 1.3.6]). By virtue of (3.12), then thesequence of matricial canonical moments associated with µ via Definition 6.1 fulfills e ( µ )0 = 1and e ( µ ) j = 1 / j ∈ N . With d := b − a let T : [0 , → [ a, b ] be defined by T ( x ) = dx + a .Then it is readily checked that ν [ a,b ] is the image measure of µ under T . Consequently, we caninfer ν [ a,b ] ∈ M < ([ a, b ]) and Z [ a,b ] t j ν [ a,b ] (d t ) = Z [0 , [ T ( x )] j µ (d x ) = Z [0 , j X k =0 jk ! d k x k a j − k µ (d x )= j X k =0 jk ! d k "Z [0 , x k µ (d x ) a j − k = j X k =0 jk ! kk ! − k d k a j − k j ∈ N . Furthermore, the sequence of matricial canonical moments associated with ν [ a,b ] coincides, according to [28, Prop. 8.12], with ( e ( µ ) j ) ∞ j =0 .We reformulate now Example 3.65(b) in the language of measures. Proposition 16.11.
Suppose δ = 2 . Denote by µ the first M [ α, β ] -transform of ν [ α,β ] . Then µ = ν [ α,β ] , i. e., the measure ν [ α,β ] is a fixed point of the M [ α, β ] -transformation. In particular,the measure ν [ − , is a fixed point of the M [ − , -transformation.Proof. Regarding Example 16.10, denote by ( s j ) ∞ j =0 the sequence of power moments associatedwith ν [ α,β ] . According to Proposition 3.7, then ( s j ) ∞ j =0 ∈ F < , ∞ ,α,β . Denote by ( t j ) ∞ j =0 the F α,β -transform of ( s j ) ∞ j =0 and by ( e j ) ∞ j =0 the [ α, β ]-interval parameter sequence of ( s j ) ∞ j =0 givenin Definitions 3.51 and 3.42, resp. Taking into account Remark 3.56 and Definition 6.5, we inferthat ( t j ) ∞ j =0 is the sequence of power moments associated with µ . By virtue of Definition 6.1,we see that ( e j ) ∞ j =0 is the sequence of matricial canonical moments associated with ν [ α,β ] . FromExample 16.10 we thus obtain e = 1 and e j = 1 / j ∈ N . Using Example 3.65(b),we can conclude then that ( t j ) ∞ j =0 coincides with ( s j ) ∞ j =0 . The application of Proposition 3.7hence yields µ = ν [ α,β ] .The following result indicates that the notion of [ α, β ]-centrality of order k of matrix mea-sures is intimately connected via Stieltjes transform with the scalar probability measure ν [ α,β ] introduced in Notation 16.9. More precisely, this property is characterized by the fact thatthe ( k − M [ α, β ]-transform of the matrix measure under consideration is a q -dimensionalinflation of ν [ α,β ] , where the corresponding matrix coefficient is a multiple of the ( k − Theorem 16.12 (cf. [29, Thm. 10.9]) . Let σ ∈ M < q ([ α, β ]) and let k ∈ N . Denote by ( d ( σ ) j ) ∞ j =0 the sequence of matricial interval lengths associated with σ given in Definition 6.1 and by σ { k − } the ( k − -th M [ α, β ] -transform of σ given via Definition 6.5. Let M := δ k − d ( σ ) k − and let µ : B [ α,β ] → C q × q be defined by µ ( B ) := [ ν [ α,β ] ( B )] M . Then σ is central of order k ifand only if σ { k − } = µ . A closer look at the proof of Theorem 16.12 given in [29] shows that one of the central pointsof it is [29, Example 10.8], where we took from [12, Example 1.3.6] the observation that thesequence ( p k ) ∞ k =1 of canonical moments of ν [0 , is the constant sequence with value 1 /
2. Thisresult originates in Karlin/Shapley [36, Sec. 25]. For an updated presentation, we refer also toKarlin/Studden [37, Ch. 4, § 4]. The essential method used by Karlin and Shapley is a carefulstudy of the geometry of Chebychev polynomials.
Example 16.13.
Let a, b ∈ R with a < b . Then ν [ a,b ] is central of order 1.Indeed, let σ := ν [ a,b ] and let M := ( b − a ) − d ( σ )0 . By virtue of Example 16.10, we see σ ∈ M < ([ a, b ]) and σ ([ a, b ]) = 1. According to Remark 6.6, we have σ { } = σ . Proposition 6.12yields σ { } ([ a, b ]) = ( b − a ) − d ( σ )0 . Consequently, we can infer M = 1 and hence σ { } ( B ) =[ ν [ a,b ] ( B )] M for all B ∈ B [ a,b ] follows. Applying Theorem 16.12 shows that ν [ a,b ] is central oforder 1.Now we turn our attention via [ α, β ]-Stieltjes transform to functions belonging to R q ( C \ [ α, β ]). 80 efinition 16.14. Let F ∈ R q ( C \ [ α, β ]) with R [ α, β ]-measure σ and let k ∈ N . We call F central of order k if σ is central of order k . Notation . Let a, b ∈ R with a < b . Then denote by g [ a,b ] the [ a, b ]-Stieltjes transform of ν [ a,b ] .The following observation is an easy consequence of the construction of the objects underconsideration. Remark . Let M ∈ C q × q < and let ν ∈ M < ([ α, β ]) with [ α, β ]-Stieltjes transform f . Then µ : B [ α,β ] → C q × q defined by µ ( B ) := [ ν ( B )] M belongs to M < q ([ α, β ]) and G : C \ [ α, β ] → C q × q defined by G ( z ) := f ( z ) M coincides with the [ α, β ]-Stieltjes transform of µ .Now we translate Theorem 16.12 into the language of functions belonging to R q ( C \ [ α, β ]). Theorem 16.17.
Let F ∈ R q ( C \ [ α, β ]) and let k ∈ N . Denote by ( d [ F ] j ) ∞ j =0 the sequence of R [ α, β ] -interval lengths associated with F given via Definition 6.3 and by F { k − } the ( k − -th R [ α, β ] -Schur transform of F given in Definition 6.8. Let N := δ k − d [ F ] k − and let G : C \ [ α, β ] → C q × q be defined by G ( z ) := g [ α,β ] ( z ) N . Then F is central of order k if and onlyif F { k − } = G .Proof. Let σ be the R [ α, β ]-measure of F . Denote by ( d ( σ ) j ) ∞ j =0 the sequence of matricialinterval lengths associated with σ and by σ { k − } the ( k − M [ α, β ]-transform of σ . Let M := δ k − d ( σ ) k − and let µ : B [ α,β ] → C q × q be defined by µ ( B ) := [ ν [ α,β ] ( B )] M . Accordingto Definitions 6.3 and 6.1, we have d [ F ] k − = d ( σ ) k − , implying N = M . From Proposition 6.12we can infer M = σ { k − } ([ α, β ]). In particular, M ∈ C q × q < . Taking additionally into accountExample 16.10 and Notation 16.15, the application of Remark 16.16 now shows that µ belongsto M < q ([ α, β ]) and that G coincides with the [ α, β ]-Stieltjes transform of µ . Consequently, inview of Definition 6.8, we can conclude from Proposition 5.5 that σ { k − } = µ if and only if F { k − } = G . By virtue of Definition 16.14, the application of Theorem 16.12 completes theproof.Theorems 16.12 and 16.17 contain further results, which indicate the importance of the arc-sine distribution introduced in Notation 16.9. For other topics in which the arcsine distributionplays a significant role, we refer to sum rules for Jacobi matrices, which are compact perturba-tions of the free Jacobi matrix associated with the arcsine distribution (see Killip/Simon [40]and Simon [52]) and free probability and random matrices (see Hiai/Petz [35])Now we are going to determine the [ α, β ]-central function associated with a sequence( s j ) mj =0 ∈ F < q,m,α,β within the description of R q [[ α, β ]; ( s j ) mj =0 ] given in Theorem 14.2. Notation . Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ]-interval lengths( d j ) mj =0 . Then let ˚ G m , ˚ X m , ˚ Y m : C \ [ α, β ] → C q × q be defined by ˚ G m ( z ) := δ m − g [ α,β ] ( z ) d m ,˚ X m ( z ) := δ m − [( β − z ) g [ α,β ] ( z ) − d m , and ˚ Y m ( z ) := ( β − z )[( z − α ) g [ α,β ] ( z )+1] P R ( d m ) + δ P N ( d m ) . Proposition 16.19.
Let m ∈ N , let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ] -interval lengths ( d j ) mj =0 . Then:(a) ˚ G m ∈ R q [[ α, β ]; ( s { m } j ) j =0 ] and [ ˚ X m ; ˚ Y m ] is the F α,β ( s { m } ) -transformed pair of ˚ G m . b) [ ˚ X m ; ˚ Y m ] ∈ ¨ P [ s { m } ] and the inverse F α,β ( s { m } ) -transform of [ ˚ X m ; ˚ Y m ] coincides with ˚ G m .Proof. Setting M := δ m − d m , we have ˚ G m = g [ α,β ] M , according to Notation 16.18. From δ > M ∈ C q × q < . Remark 14.3 furthermore yields s { m } = M .Denote by [ G ; G ] the F α,β ( M )-transformed pair of ˚ G m .(a) Taking into account Example 16.10 and Notation 16.15, the application of Remark 16.16shows that µ : B [ α,β ] → C q × q defined by µ ( B ) := [ ν [ α,β ] ( B )] M belongs to M < q ([ α, β ]) andthat ˚ G m coincides with the [ α, β ]-Stieltjes transform of µ . In view of Example 16.10, we have µ ([ α, β ]) = M . Consequently, µ ∈ M < q, [[ α, β ]; ( s { m } j ) j =0 , =] follows. Remark 5.7 then shows˚ G m ∈ R q [[ α, β ]; ( s { m } j ) j =0 ]. Taking into account M ∗ = M , we can infer from Remark 14.3moreover P R ( M ∗ ) = P R ( d m ) and P N ( M ) = P N ( d m ) . Using Remark A.18, in particular M † M = P R ( d m ) follows. By virtue of Definition 9.1 and Notation 16.18 we have, for all z ∈ C \ [ α, β ],then G ( z ) = ( β − z ) ˚ G m ( z ) − M = ( β − z ) g [ α,β ] ( z ) M − M = h ( β − z ) g [ α,β ] ( z ) − i M = ˚ X m ( z )and G ( z ) = ( β − z ) h ( z − α ) M † ˚ G m ( z ) + P R ( M ∗ ) i + δ P N ( M ) = ( β − z ) h ( z − α ) g [ α,β ] ( z ) M † M + P R ( M ∗ ) i + δ P N ( M ) = ( β − z ) h ( z − α ) g [ α,β ] ( z ) P R ( d m ) + P R ( d m ) i + δ P N ( d m ) = ( β − z ) h ( z − α ) g [ α,β ] ( z ) + 1 i P R ( d m ) + δ P N ( d m ) = ˚ Y m ( z ) . (b) By virtue of Proposition 3.58, we have ( s { m } j ) j =0 ∈ F < q, ,α,β . Taking additionallyinto account part (a), Lemma 11.1 yields [ ˚ X m ; ˚ Y m ] ∈ ¨ P [ s { m } ]. Obviously, P R ( M ) ˚ G m = g [ α,β ] P R ( M ) M = ˚ G m . Because of M ∈ C q × q < and part (a), the application of Lemma 9.14shows that the inverse F α,β ( M )-transform of [ G ; G ] coincides with ˚ G m . Proposition 16.20.
Let m ∈ N , let ( s j ) mj =0 ∈ F < q,m,α,β with sequence of [ α, β ] -interval lengths ( d j ) mj =0 , and let F ∈ R q [[ α, β ]; ( s j ) mj =0 ] . Denote by ¨ G m ( F, ( s j ) mj =0 ) the m -th F α,β -transformof F with respect to ( s j ) mj =0 and by P ¨ G m ( F, ( s j ) mj =0 ) the m -th F α,β -transformed pair of F with respect to ( s j ) mj =0 given in Definition 13.1. Regarding Notation 16.18 and Definition 7.11,then the following statements are equivalent:(i) F is central of order m + 1 .(ii) ¨ G m ( F, ( s j ) mj =0 ) = ˚ G m .(iii) P ¨ G m ( F, ( s j ) mj =0 ) ∼ [ ˚ X m ; ˚ Y m ] .Proof. Denote by F { m } the m -th R [ α, β ]-Schur transform of F . Due to Proposition 13.8, wehave ¨ G m ( F, ( s j ) mj =0 ) = F { m } .(i) ⇔ (ii): Denote by σ the R [ α, β ]-measure of F and by ( s ( σ ) j ) ∞ j =0 and ( d ( σ ) j ) ∞ j =0 the sequenceof power moments and the sequence of matricial interval lengths associated with σ , resp. Then82 ∈ M < q,m [[ α, β ]; ( s j ) mj =0 , =] and hence s ( σ ) j = s j for all j ∈ Z ,m . By virtue of Remark 3.22,consequently d ( σ ) m = d m . Denote by ( d [ F ] j ) ∞ j =0 the sequence of R [ α, β ]-interval lengths associatedwith F . Taking additionally into account Definitions 6.3 and 6.1, then d [ F ] m = d ( σ ) m = d m follows.According to Notation 16.18, thus ˚ G m = g [ α,β ] δ m − d [ F ] m . Now, in view of ¨ G m ( F, ( s j ) mj =0 ) = F { m } , the application of Theorem 16.17 yields the equivalence of (i) and (ii).(ii) ⇒ (iii): From Proposition 16.19(a) we see that [ ˚ X m ; ˚ Y m ] is the F α,β ( s { m } )-transformedpair of ˚ G m . In view of (ii) and Definition 13.1, then P ¨ G m ( F, ( s j ) mj =0 ) = [ ˚ X m ; ˚ Y m ] follows. Inparticular, (iii) holds true.(iii) ⇒ (ii): Setting M := s { m } , the combination of Proposition 3.58 and Lemma 3.11yields M ∈ C q × q < . From Proposition 16.19(b) we see [ ˚ X m ; ˚ Y m ] ∈ ¨ P [ M ] and that ˚ G m is the inverse F α,β ( M )-transform of [ ˚ X m ; ˚ Y m ]. Because of (iii) and Remark 8.4, in par-ticular P ¨ G m ( F, ( s j ) mj =0 ) ∈ ¨ P [ M ] follows. Denote by G the inverse F α,β ( M )-transform of P ¨ G m ( F, ( s j ) mj =0 ). Taking into account (iii), we can infer from Corollary 9.12 that G = ˚ G m .Observe that P ¨ G m ( F, ( s j ) mj =0 ) is the F α,β ( M )-transformed pair of ¨ G m ( F, ( s j ) mj =0 ), accord-ing to Definition 13.1. Remark 6.11 yields F { m } ∈ R q [[ α, β ]; ( s { m } j ) j =0 ]. Consequently, the R [ α, β ]-measure µ of F { m } belongs to M < q, [[ α, β ]; ( s { m } j ) j =0 , =], i. e., µ ([ α, β ]) = M . Takingadditionally into account Proposition 4.15(a), hence R ( F { m } ( z )) = R ( M ) for all z ∈ C \ [ α, β ]follows. In view of ¨ G m ( F, ( s j ) mj =0 ) = F { m } , hence P R ( M ) ¨ G m ( F, ( s j ) mj =0 ) = ¨ G m ( F, ( s j ) mj =0 ).Thus, we can apply Lemma 9.14 to obtain G = ¨ G m ( F, ( s j ) mj =0 ). Therefore, (ii) holds true.Now we are able to determine the [ α, β ]-Stieltjes transform of the [ α, β ]-central measureassociated with a sequence ( s j ) mj =0 ∈ F < q,m,α,β . Proposition 16.21.
Let m ∈ N and let ( s j ) mj =0 ∈ F < q,m,α,β . Denote by h ˜ w m ˜ x m ˜ y m ˜ z m i the q × q blockrepresentation of the restriction of ¨ V m onto C \ [ α, β ] . Then det(˜ y m ˚ X m +˜ z m ˚ Y m ) does not vanishidentically in C \ [ α, β ] and ˚ S m = ( ˜ w m ˚ X m + ˜ x m ˚ Y m )(˜ y m ˚ X m + ˜ z m ˚ Y m ) − .Proof. Remark 16.19(b) shows [ ˚ X m ; ˚ Y m ] ∈ ¨ P [ s { m } ]. Consequently, we can apply Theo-rem 14.2(a) to see that det(˜ y m ˚ X m +˜ z m ˚ Y m ) does not vanish identically in C \ [ α, β ] and that thematrix-valued function F := ( ˜ w m ˚ X m + ˜ x m ˚ Y m )(˜ y m ˚ X m + ˜ z m ˚ Y m ) − belongs to R q [[ α, β ]; ( s j ) mj =0 ].Theorem 14.2(b) then yields [ ˚ X m ; ˚ Y m ] ∈ h P ¨ G m ( F, ( s j ) mj =0 ) i . From Proposition 16.20 we canthus conclude that F is central of order m + 1. Denote by σ the R [ α, β ]-measure of F . Then σ ∈ M < q,m [[ α, β ]; ( s j ) mj =0 , =]. According to Definition 16.14, furthermore σ is central of order m + 1. In view of Definition 16.8, this means that the sequence of power moments ( s ( σ ) j ) ∞ j =0 associated with σ is [ α, β ]-central of order m + 1. Since s ( σ ) j = s j for all j ∈ Z ,m , thus Def-initions 16.2 and 16.3 show that ( s ( σ ) j ) ∞ j =0 coincides with the [ α, β ]-central sequence (˚ s j ) ∞ j =0 associated with ( s j ) mj =0 . Consequently, σ ∈ M < q, ∞ [[ α, β ]; (˚ s j ) ∞ j =0 , =] follows. Proposition 16.5and Definition 16.6 then yield σ = ˚ σ m . From Proposition 5.5 we can furthermore conclude that F is the [ α, β ]-Stieltjes transform of σ . In view of Definition 16.7, the proof is complete.In our following considerations, we concentrate on the case of a sequence ( s j ) mj =0 ∈ F ≻ q,m,α,β .Before doing that we state some elementary preparations in the scalar case, which are of owninterest. 83 emma 16.22. Let a, b ∈ R with a < b and let z ∈ C \ [ a, b ] . Then there exists a unique w ∈ C satisfying w = ( z − a )( z − b ) and | w − z + c | < d , where c := ( a + b ) / and let d := ( b − a ) / .Proof. First observe that ( z − a )( z − b ) = ( z − c ) − d and that there exists either a singleone or two different solutions w ∈ C satisfying w = ( z − a )( z − b ). We choose a particularsolution w . If w = 0, then z = a or z = b , contradicting z / ∈ [ a, b ]. Thus, we have w = − w and hence w := w and w := − w are the only solutions of the equation w = ( z − a )( z − b ).Consequently, t := w − z + c and t := w − z + c fulfill t = t and ( t , + z − c ) = ( ± w ) =( z − c ) − d . Therefore, t and t are the two solutions of the equation t + 2( z − c ) t + d = 0.Hence, t + t = − z − c ) and t t = d . In particular, | t |·| t | = d . We are now going to show | t | 6 = | t | . Assume to the contrary | t | = | t | . In view of t t = d and d >
0, then t = t and | t | = d follow. Using t + t = − z − c ), we can thus infer z = c − Re t ∈ R and furthermore | z − c | = | Re t | ≤ | t | . Taking additionally into account | t | = d , then − d ≤ z − c ≤ d follows,contradicting z / ∈ [ a, b ]. Thus, we have shown | t | 6 = | t | . Since | t | · | t | = d , then either | t | < d and | t | > d or | t | > d and | t | < d . Consequently, exactly one of the two solutions ofthe equation w = ( z − a )( z − b ) fulfills | w − z + c | < d . Lemma 16.23. (a) The function g [ − , belongs to R ( C \ [ − , with R [ − , -measure ν [ − , and is central of order .(b) Let z ∈ C \ [ − , . Then g [ − , ( z ) = 0 and w z := − /g [ − , ( z ) is the unique complexnumber w satisfying w = z − and | w − z | < .Proof. (a) In view of Example 16.10 and Notation 16.15, we can infer from Proposition 5.5 that g [ − , belongs to R ( C \ [ − , ν [ − , is the R [ − , g [ − , . Accordingto Definition 16.14 and Example 16.13, thus g [ − , is central of order 1.(b) Because of part (a), we can apply Proposition 4.15(a) to obtain N ( g [ − , ( z )) = N ( ν [ − , ([ − , ν [ − , ([ − , g [ − , ( z ) = 0follows. According to [12, p. 125, especially formula (4.5.4)], the remaining assertion ofpart (b) holds true. (Observe that in [12], for probability measures µ on [ − , S ( z, µ ) = R − ( z − x ) − d µ ( x ) = − ¨ S µ ( z ) is considered.) Proposition 16.24. (a) The function g [ α,β ] belongs to R ( C \ [ α, β ]) with R [ α, β ] -measure ν [ α,β ] and is central of order .(b) Let z ∈ C \ [ α, β ] . Then g [ α,β ] ( z ) = 0 and w z := − /g [ α,β ] ( z ) is the unique complexnumber w satisfying w = ( z − α )( z − β ) and | w − z + ( α + β ) / | < ( β − α ) / .Proof. By the same reasoning as in the proof of Lemma 16.23, we can conclude that g [ α,β ] belongs to R ( C \ [ α, β ]) and is central of order 1, that ν [ α,β ] is the R [ α, β ]-measure of g [ α,β ] ,and that g [ α,β ] ( z ) = 0. Let c := ( α + β ) / d := ( β − α ) /
2. Let T : [ − , → [ α, β ] bedefined by T ( x ) = dx + c . Then it is readily checked that ν [ α,β ] is the image measure of ν [ − , under T . Observe that d > z / ∈ [ α, β ] implies ζ / ∈ [ − ,
1] for ζ := ( z − c ) /d . Byvirtue of Notation 16.15 and Definition 5.3, we thus can infer g [ α,β ] ( z ) = Z [ α,β ] t − z ν [ α,β ] (d t ) = Z [ − , T ( x ) − z ν [ − , (d x ) = Z [ − , dx − z + c ν [ − , (d x )= 1 d Z [ − , x − ( z − c ) /d ν [ − , (d x ) = 1 d g [ − , ( ζ ) .
84n view of Lemma 16.23(b), consequently w z = dω ζ , where ω ζ is the unique complex number ω satisfying ω = ζ − | ω − ζ | <
1. Hence, w z = d ( ζ −
1) = ( z − c ) − d = ( z − α )( z − β )and | w z − z + ( α + β ) / | = | dω ζ − z + c | = d | ω ζ − ζ | < d = ( β − α ) /
2. By virtue of Lemma 16.22,the proof is complete.
Lemma 16.25.
Suppose δ = 2 . Denote by f the first R [ α, β ] -Schur transform of g [ α,β ] .Then f = g [ α,β ] , i. e., the function g [ α,β ] is a fixed point of the R [ α, β ] -Schur transformation.In particular, the function g [ − , is a fixed point of the R [ − , -Schur transformation.Proof. In view of Example 16.10 and Notation 16.15, we can infer from Proposition 5.5 that g [ α,β ] belongs to R ( C \ [ α, β ]) and that ν [ α,β ] is the R [ α, β ]-measure of g [ α,β ] . Denote by µ thefirst M [ α, β ]-transform of ν [ α,β ] . According to Definition 6.8, then f is the [ α, β ]-Stieltjes trans-form of µ . Since Proposition 16.11 yields µ = ν [ α,β ] , hence f is the [ α, β ]-Stieltjes transform of ν [ α,β ] . Regarding Notation 16.15, the proof is complete. Proposition 16.26.
Let m ∈ N and let ( s j ) mj =0 ∈ F ≻ q,m,α,β with sequence of [ α, β ] -intervallengths ( d j ) mj =0 . Let G , G : C \ [ α, β ] → C q × q be defined by G ( z ) := δ m − g [ α,β ] ( z ) d m and G ( z ) := I q . For all z ∈ C \ [ α, β ] , then ˚ X m ( z ) = δ m − h ( β − z ) g [ α,β ] ( z ) − i d m , ˚ Y m ( z ) = ( β − z ) h ( z − α ) g [ α,β ] ( z ) + 1 i I q . (16.1) Regarding Definition 7.11, furthermore [ G ; G ] ∈ PR q ( C \ [ α, β ]) and [ ˚ X m ; ˚ Y m ] ∼ [ G ; G ] .Proof. Remark 3.66 yields det d m = 0. Hence, R ( d m ) = C q and N ( d m ) = { O q × } , whichimply P R ( d m ) = I q and P N ( d m ) = O q × q . For all z ∈ C \ [ α, β ], now (16.1) follows imme-diately from Notation 16.18. By virtue of Proposition 16.19(b) and Notation 8.1, we get[ ˚ X m ; ˚ Y m ] ∈ PR q ( C \ [ α, β ]). Let f : C \ [ α, β ] → C be defined by f ( z ) := ( z − α ) g [ α,β ] ( z ) + 1.From Proposition 16.24(a) we see that g [ α,β ] belongs to R q ( C \ [ α, β ]) and that ν [ α,β ] isthe R [ α, β ]-measure of g [ α,β ] . Example 16.10 furthermore shows ν [ α,β ] ([ α, β ]) = 1 and R [ α,β ] tν [ α,β ] (d t ) = ( α + β ) /
2. Taking into account Notation 4.19, we can thus infer fromProposition 4.20 that f belongs to R ( C \ [ α, β ]) and that the R [ α, β ]-measure σ of f fulfills σ ([ α, β ]) = R [ α,β ] ( t − α ) ν [ α,β ] (d t ) = ( α + β ) / − α = ( β − α ) / = 0. Consider now an arbi-trary z ∈ C \ [ α, β ]. Using Proposition 4.15(a) we obtain then N ( f ( z )) = N ( σ ([ α, β ])) = { } and hence f ( z ) = 0. Regarding ˚ Y m ( z ) = ( β − z ) f ( z ) I q , then det ˚ Y m ( z ) = 0 fol-lows. From Proposition 16.24(b) we see g [ α,β ] ( z ) = 0 and that w z := − /g [ α,β ] ( z ) satis-fies w z = ( z − α )( z − β ). Hence, [( β − z ) + w z ] w z = ( β − z )[ w z − ( z − α )] and, in viewof w z = 0, therefore ( β − z ) + w z = ( β − z )[1 − ( z − α ) /w z ] = ( β − z ) f ( z ). Multipli-cation by g [ α,β ] ( z ) yields ( β − z ) g [ α,β ] ( z ) − β − z ) f ( z ) g [ α,β ] ( z ). Consequently, we getthe identity [( β − z ) g [ α,β ] ( z ) − / [( β − z ) f ( z )] = g [ α,β ] ( z ), from which we can conclude[ ˚ X m ( z )][˚ Y m ( z )] − = G ( z ). The application of Lemma 7.13 completes the proof. Proposition 16.27.
Let m ∈ N and let ( s j ) mj =0 ∈ F ≻ q,m,α,β . Let h ˜ w m ˜ x m ˜ y m ˜ z m i be the q × q blockrepresentation of the restriction of ¨ V m onto C \ [ α, β ] . Then det( δ m − g [ α,β ] ˜ y m d m + ˜ z m ) doesnot vanish identically in C \ [ α, β ] and ˚ S m = ( δ m − g [ α,β ] ˜ w m d m + ˜ x m )( δ m − g [ α,β ] ˜ y m d m + ˜ z m ) − .Proof. From Proposition 16.21 we see that det(˜ y m ˚ X m + ˜ z m ˚ Y m ) does not vanish identically in C \ [ α, β ] and that ˚ S m = ( ˜ w m ˚ X m +˜ x m ˚ Y m )(˜ y m ˚ X m +˜ z m ˚ Y m ) − . Remark 16.19(b) shows [ ˚ X m ; ˚ Y m ] ∈ ¨ P [ s { m } ]. Let G , G : C \ [ α, β ] → C q × q be defined by G ( z ) := δ m − g [ α,β ] ( z ) d m and G ( z ) := I q .85ccording to Proposition 16.26, then [ G ; G ] ∈ PR q ( C \ [ α, β ]) and [ ˚ X m ; ˚ Y m ] ∼ [ G ; G ]. Theapplication of Definition 7.11 completes the proof. A. Some facts from matrix theory
This appendix contains a summary of results from matrix theory, which are used in this paper.What concerns results on the Moore–Penrose inverse A † of a complex matrix A , we refer, e. g.,to [13, Sec. 1]. Remark
A.1 . Let m, n ∈ N and let A , A , . . . , A m and B , B , . . . , B n be complex p × q ma-trices. If M ∈ C q × p is such that A j M B k = B k M A j holds true for all j ∈ Z ,m and all k ∈ Z ,n , then ( P mj =1 η j A j ) M ( P nk =1 θ k B k ) = ( P nk =1 θ k B k ) M ( P mj =1 η j A j ) for all complex num-bers η , η , . . . , η m and θ , θ , . . . , θ n . Remark
A.2 . (a) If Z ∈ C q × q and η ∈ C , then Re( ηZ ) = Re( η ) Re( Z ) − Im( η ) Im( Z ) andIm( ηZ ) = Re( η ) Im( Z ) + Im( η ) Re( Z ).(b) If Z ∈ C q × q and X ∈ C q × p , then Re( X ∗ ZX ) = X ∗ Re( Z ) X and Im( X ∗ ZX ) = X ∗ Im( Z ) X . Remark
A.3 . If A ∈ C p × q , then dim R ( A ) + dim N ( A ) = q . Remark
A.4 . If A ∈ C p × q , then R ( A ) = R ( AA ∗ ) and N ( A ) = N ( A ∗ A ). Remark
A.5 . Let A ∈ C p × r , let B ∈ C p × s , and let C ∈ C q × r . In view of Remark A.4, then R ( A ) + R ( B ) = R ([ A, B ]) = R ( AA ∗ + BB ∗ ) and N ( A ) ∩ N ( C ) = N ( (cid:2) AC (cid:3) ) = N ( A ∗ A + C ∗ C ). Remark
A.6 . Let n ∈ N and let A , A , . . . , A n ∈ C p × q . For all η , η , . . . , η n ∈ C , then R ( P nj =1 η j A j ) ⊆ P nj =1 R ( A j ) and T nj =1 N ( A j ) ⊆ N ( P nj =1 η j A j ). Remark
A.7 . If A ∈ C p × q has rank q and B ∈ C q × s has rank s , then rank( AB ) = s . Remark
A.8 . Let L ∈ C p × p and R ∈ C q × q be both invertible. Let A ∈ C p × q and let X := LAR .Then R ( X ) = L R ( A ) and N ( X ) = R − N ( A ).We think that the following result is well-know. However, we did not succeed in finding areference. Lemma A.9.
Let A ∈ C p × q and let R ∈ C q × r . Then R ( AR ) = R ( A ) if and only if N ( A ) + R ( R ) = C q .Proof. Observe that U := N ( A ) + R ( R ) is a linear subspace of the C -vector space C q . Let φ : U → C p be defined by φ ( x ) := Ax . Then φ is linear with ker φ = U ∩ N ( A ) = N ( A ) and φ ( U ) = A R ( R ) = R ( AR ). Regarding dim ker φ +dim φ ( U ) = dim U , then dim U = dim N ( A )+dim R ( AR ) = q − rank A + rank( AR ) follows. Hence, U = C q if and only if rank( AR ) = rank A .Because of R ( AR ) ⊆ R ( A ), the latter is equivalent to R ( AR ) = R ( A ). Remark
A.10 . If A ∈ C p × q , then R ( A ∗ ) = [ N ( A )] ⊥ and N ( A ∗ ) = [ R ( A )] ⊥ .We write P U for the transformation matrix corresponding to the orthogonal projection ontoa linear subspace U of the unitary space C p with respect to the standard basis. Remark
A.11 . Let U be a linear subspace of C p . Then P U is the unique complex p × p matrixsatisfying P U = P U , P ∗U = P U , and R ( P U ) = U . Furthermore, N ( P U ) = U ⊥ and P U + P U ⊥ = I p . Remark
A.12 . If U is a linear subspace of the unitary space C p with dimension d := dim U ≥ u , u , . . . , u d , then P U = U U ∗ , where U := [ u , u , . . . , u d ].86 emark A.13 . If A ∈ C q × q fulfills det A = 0, then A † = A − . Remark
A.14 . If A ∈ C p × q , then ( A † ) † = A , ( A ∗ ) † = ( A † ) ∗ , R ( A † ) = R ( A ∗ ), and N ( A † ) = N ( A ∗ ). Remark
A.15 . Let η ∈ C and let A ∈ C p × q . Then ( ηA ) † = η † A † . Remark
A.16 . If A ∈ C q × q < , then A † ∈ C q × q < and ( A † ) / = ( A / ) † .Regarding (3.9), we easily obtain with Remark A.15: Remark
A.17 . Let η ∈ C and let A, B ∈ C p × q . Then ( ηA ) ⊤−⊥ ( ηB ) = η ( A ⊤−⊥ B ). Remark
A.18 . If A ∈ C p × q , then P R ( A ) = AA † , P N ( A ) = I q − A † A , P R ( A ∗ ) = A † A , and P N ( A ∗ ) = I p − AA † . Lemma A.19.
Let A ∈ C p × q and let B ∈ C q × q be such that R ( B ) ⊆ R ( A ∗ ) ⊆ R ( B ∗ ) . Let η ∈ C \ { } . Then the matrix B + η P N ( A ) is invertible and B † = ( B + η P N ( A ) ) − − η − P N ( A ) .Proof. First observe that R ( B ) = R ( A ∗ ) = R ( B ∗ ) follows from the assumption, sincerank( B ∗ ) = rank B holds true. In view of Remark A.18, we thus obtain BB † = A † A = B † B and P N ( A ) = I q − A † A . Taking additionally into account Remark A.11, we infer then( B + η P N ( A ) )( B † + η − P N ( A ) ) = BB † + η − B P N ( A ) + η P N ( A ) B † + P N ( A ) = A † A + η − B ( I q − A † A ) + η ( I q − A † A ) B † + P N ( A ) = I q + η − B ( I q − B † B ) + η ( I q − B † B ) B † = I q . Remark
A.20 . Let A ∈ C p × q and let B ∈ C p × m . Then R ( B ) ⊆ R ( A ) if and only if AA † B = B . Remark
A.21 . Let A ∈ C p × q and let C ∈ C n × q . Then N ( A ) ⊆ N ( C ) if and only if CA † A = C .The combination of Remarks A.20, A.21, and A.1 yields: Remark
A.22 . Let A ∈ C p × q and M ∈ C q × p be such that R ( A ) ⊆ R ( M ) and N ( M ) ⊆ N ( A ).For all η , η , θ , θ ∈ C , then ( η A + η M ) M † ( θ A + θ M ) = ( θ A + θ M ) M † ( η A + η M ).Regarding Remark A.20, we can easily conclude from Lemma A.9: Remark
A.23 . Let A ∈ C p × q and let B ∈ C p × r . Then R ( B ) = R ( A ) if and only if there existsa complex q × r matrix R fulfilling N ( A ) + R ( R ) = C q and B = AR . Remark
A.24 . The set C q × q H is an R -vector space and C q × q < is a convex cone in C q × q H . Remark
A.25 . Let A ∈ C q × q H and let X ∈ C q × p . Then X ∗ AX ∈ C p × p H . If A ∈ C q × q < , then X ∗ AX ∈ C p × p < .We now state a well-known characterization of non-negative Hermitian block matrices interms of the Schur complement (3.4): Lemma A.26 (cf. [13, Lem. 1.1.9 and 1.1.10]) . Let (cid:2)
A BC D (cid:3) be the block representation of acomplex ( p + q ) × ( p + q ) matrix M with p × p block A . Then M is non-negative Hermitianif and only if A and M/A := D − CA † B are both non-negative Hermitian and furthermore R ( B ) ⊆ R ( A ) and C = B ∗ are fulfilled. In this case, k B k ≤ k A k S k D k S . Lemma A.27 (cf. [28, Lem. A.13]) . Let
A, B ∈ C q × q H with O q × q A B . Then R ( A ) ⊆ R ( B ) and N ( B ) ⊆ N ( A ) . Furthermore, O q × q P R ( A ) B † P R ( A ) A † .
87e continue with some observations on the classes of EP matrices and almost definite ma-trices given in Definition 4.16. From Remark A.6 we easily see:
Remark
A.28 . If A ∈ C q × q EP , then R (Re A ) ⊆ R ( A ) and R (Im A ) ⊆ R ( A ). Remark
A.29 . If A ∈ C q × q AD , then ηA ∈ C q × q AD for all η ∈ C . Remark
A.30 . If A ∈ C q × q AD , then N ( A ∗ ) = N ( A ) and R ( A ∗ ) = R ( A ). Remark
A.31 . Taking into account Remark A.30, one can easily check that { M ∈ C q × q : ηM ∈ C q × q < } ⊆ C q × q AD ⊆ C q × q EP for all η ∈ C . Lemma A.32.
Let A ∈ C q × q satisfy Im A ∈ C q × q < . Then A ∈ C q × q EP .Proof. Let x ∈ N ( A ). Then x ∗ Im( A ) x = 0. Since, by virtue of Remark A.31, we have Im A ∈ C q × q AD , then Im( A ) x = O follows. Consequently, A ∗ x = Ax = O . Hence, N ( A ) ⊆ N ( A ∗ ),implying R ( A ) = R ( A ∗ ), i. e., A ∈ C q × q EP . Lemma A.33.
Let A ∈ C q × q satisfy Im A ∈ C q × q < and rank(Im A ) = rank A . Then A ∈ C q × q AD and R (Im A ) = R ( A ) .Proof. Lemma A.32 yields R ( A ∗ ) = R ( A ). In view of Remark A.28 and rank(Im A ) = rank A ,we get R (Im A ) = R ( A ). Consider an arbitrary x ∈ C q with x ∗ Ax = 0. Then x ∗ A ∗ x = x ∗ Ax =0. Consequently, x ∗ Im( A ) x = 0. Since Remark A.31 yields Im A ∈ C q × q AD , we have Im( A ) x = O q × . Because of N (Im A ) = [ R ((Im A ) ∗ )] ⊥ = [ R (Im A )] ⊥ = [ R ( A )] ⊥ = [ R ( A ∗ )] ⊥ = N ( A ),we get Ax = O q × and, thus, A ∈ C q × q AD .A complex p × q matrix K is said to be contractive if k K k S ≤ Remark
A.34 . Let K ∈ C p × q . Then the matrix K is contractive if and only if I q − K ∗ K isnon-negative Hermitian. Remark
A.35 . Let (cid:2)
A BC D (cid:3) be the block representation of a contractive complex( p + q ) × ( p + q ) matrix K with p × p block A . Suppose that A or D is unitary. In viewof Remark A.34 then one can easily see that B = O p × q and C = O q × p .A complex square matrix J is called a signature matrix if it satisfies J ∗ = J and J = I . Inthis paper, we focus on the particular signature matrices˜ J q := " O q × q i I q − i I q O q × q and j pq := " − I p O p × q O q × p I q . (A.1)What concerns several aspects of the so-called J -Theory for arbitrary signature matrices J , werefer to [13, §§ 1.3–1.4, p. 26–44]. Remark
A.36 . If P, Q ∈ C q × p , then (cid:2) PQ (cid:3) ∗ ˜ J q (cid:2) PQ (cid:3) = 2 Im( Q ∗ P ) and (cid:2) PQ (cid:3) ∗ j qq (cid:2) PQ (cid:3) = Q ∗ Q − P ∗ P . Remark
A.37 . Let A ∈ C p × p and let D ∈ C q × q . Then h A O p × q O q × p D i j pq = h − A O p × q O q × p D i = j pq h A O p × q O q × p D i . Furthermore, if p = q , then h A O q × q O q × p D i ˜ J q = h O q × q i A − i D O q × q i = ˜ J q h D O q × q O q × p A i . Inparticular, ˜ J q j qq = − j qq ˜ J q . 88 . Integration with respect to non-negative Hermitian measures Consider a measurable space ( X , X ) consisting of a non-empty set X and a σ -algebra X on X . A mapping µ : X → C q × q < is called non-negative Hermitian q × q measure on ( X , X ) if itis σ -additive, i. e. µ ( S ∞ n =1 X n ) = P ∞ n =1 µ ( X n ) holds true for all sequences ( X n ) ∞ n =1 of pairwisedisjoint sets from X . We are using the integration theory of pairs of matrix-valued functionswith respect to non-negative Hermitian measures, developed independently by Kats [38] andRosenberg [47–49](cf. [13, Sec. 2.2]):First consider an arbitrary ordinary measure ν on a measurable space ( X , X ). A measurablefunction F : X → C p × q is said to be integrable with respect to ν if F = [ f jk ] j =1 ,...,pk =1 ,...,q belongsto [ L ( ν )] p × q , i. e., all entries f jk belong to the class L ( ν ) of functions f : X → C , which areintegrable with respect to ν . In this case, let R X F d ν := [ R X f jk d ν ] j =1 ,...,pk =1 ,...,q for all X ∈ X .Throughout this part of the appendix, let an arbitrary non-negative Hermitian q × q measure µ = [ µ jk ] qj,k =1 be given. Then all the entries µ jk are complex-valued measures and all thediagonal entries µ jj are ordinary measures with values in [0 , ∞ ). Consequently, the tracemeasure τ := tr µ = P qj =1 µ jj of µ is an ordinary measure with values in [0 , ∞ ). For each X ∈ X ,from τ ( X ) = 0 necessarily µ ( X ) = O q × q follows. Consequently, there exist τ -a. e. uniquelydetermined Radon–Nikodym derivatives d µ jk / d τ . The τ -a. e. uniquely determined measurablemapping µ ′ τ := [d µ jk / d τ ] qj,k =1 is called trace derivative of µ and satisfies µ ( X ) = R X µ ′ τ d τ forall X ∈ X and O q × q µ ′ τ ( x ) I q for τ -a. a. x ∈ X .A measurable function f : X → C is said to be integrable with respect to µ if R X | f | d ν jk < ∞ holds true for all j, k ∈ Z ,q , where ν jk denotes the variation of the complex measure µ jk . Inthis case, let R X f d µ := [ R X f d µ jk ] qj,k =1 for all X ∈ X . Denote by L ( µ ) the set of all suchfunctions f , which are integrable with respect to µ in this sense. Remark
B.1 . Let u ∈ C q and let ν := u ∗ µu . Then ν is a bounded measure on ( X , X ), whichis absolutely continuous with respect to τ . For all f ∈ L ( µ ), furthermore R X | f | d ν < ∞ and R X f d ν = u ∗ ( R X f d µ ) u . Remark
B.2 . The mapping defined on the C -vector space L ( µ ) by f R X f d µ is C -linear. Remark
B.3 . If f ∈ L ( µ ), then f ∈ L ( µ ) and R X f d µ = ( R X f d µ ) ∗ .An ordered pair (Φ , Ψ) consisting of measurable functions Φ :
X → C p × q and Ψ : X → C r × q is said to be left-integrable with respect to µ if the matrix-valued function Φ µ ′ τ Ψ ∗ belongs to[ L ( τ )] p × r . In this case, let R X Φd µ Ψ ∗ := R X Φ µ ′ τ Ψ ∗ d τ for all X ∈ X . In particular, denoteby L p × q ( µ ) the set of all measurable functions Φ : X → C p × q for which the pair (Φ , Φ) isleft-integrable with respect to µ . Remark
B.4 . If Φ ∈ L p × q ( µ ) and Ψ ∈ L r × q ( µ ), then (Φ , Ψ) is left-integrable with respect to µ . Remark
B.5 . If Φ , Ψ ∈ L p × q ( µ ), then R X Φd µ Ψ ∗ = ( R X Ψd µ Φ ∗ ) ∗ and R X Φd µ Φ ∗ ∈ C p × p < for all X ∈ X . Lemma B.6.
Let Φ ∈ L p × q ( µ ) and let Ψ ∈ L r × q ( µ ) . For all X ∈ X , then R (cid:18)Z X Φd µ Ψ ∗ (cid:19) ⊆ R (cid:18)Z X Φd µ Φ ∗ (cid:19) , N (cid:18)Z X Φd µ Φ ∗ (cid:19) ⊆ N (cid:18)Z X Ψd µ Φ ∗ (cid:19) , and (cid:18)Z X Ψd µ Φ ∗ (cid:19)(cid:18)Z X Φd µ Φ ∗ (cid:19) † (cid:18)Z X Φd µ Ψ ∗ (cid:19) Z X Ψd µ Ψ ∗ . roof. Consider an arbitrary X ∈ X . First observe that Θ := (cid:2) ΦΨ (cid:3) is a measurable functionsatisfying Θ µ ′ τ Θ ∗ = " ΦΨ µ ′ τ [Φ ∗ , Ψ ∗ ] = " Φ µ ′ τ Φ ∗ Φ µ τ Ψ ∗ Ψ µ τ Φ ∗ Ψ µ ′ τ Ψ ∗ . By assumption, we have Φ µ ′ τ Φ ∗ ∈ [ L ( τ )] p × p and Ψ µ ′ τ Ψ ∗ ∈ [ L ( τ )] r × r . Due to Remark B.4, thepairs (Φ , Ψ) and (Ψ , Φ) are both left-integrable with respect to µ , i. e., Φ µ τ Ψ ∗ ∈ [ L ( τ )] p × r andΨ µ τ Φ ∗ ∈ [ L ( τ )] r × p are true. Thus, we infer Θ µ ′ τ Θ ∈ [ L ( τ )] ( p + r ) × ( p + r ) , i. e., Θ ∈ L p + r ) × q ( µ ).Setting A := R X Φd µ Φ ∗ , B := R X Φd µ Ψ ∗ , C := R X Ψd µ Φ ∗ , and D := R X Ψd µ Ψ ∗ , we get " A BC D = Z X " Φ µ ′ τ Φ ∗ Φ µ ′ τ Ψ ∗ Ψ µ ′ τ Φ ∗ Ψ µ ′ τ Ψ ∗ d τ = Z X Θ µ ′ τ Θ ∗ d τ = Z X Θd µ Θ ∗ . Because of Remark B.5, the matrix R X Θd µ Θ ∗ is non-negative Hermitian. Using Lemma A.26,we can conclude then R ( B ) ⊆ R ( A ), C = B ∗ , and that the matrices A , D , and E/A = D − CA † B are non-negative Hermitian. By virtue of Remark A.14, thus D and CA † B areHermitian matrices, which satisfy CA † B D . Taking into account Remark A.10, we canfurthermore infer N ( A ) ⊆ N ( C ). Remark
B.7 . Let f : X → C be a measurable function and let Φ , Ψ :
X → C q × q be defined byΦ( x ) := f ( x ) I q and Ψ( x ) := I q . Then f ∈ L ( µ ) if and only if (Φ , Ψ) is left-integrable withrespect to µ . In this case, f µ ′ τ ∈ [ L ( τ )] q × q and R X f d µ = R X ( f µ ′ τ )d τ = R X Φd µ Ψ ∗ for all X ∈ X , where µ ′ τ denotes the trace derivative of µ .Using Remark B.7 and Lemma B.6, the following result can be easily verified: Lemma B.8 (cf. [22, Lem. B.2(b)]) . If f ∈ L ( µ ) , then R ( R X f d µ ) ⊆ R ( µ ( X )) and N ( µ ( X )) ⊆N ( R X f d µ ) . Lemma B.9.
Let g : X → R satisfy g ∈ L ( µ ) and µ ( { g ≤ } ) = O q × q . Then R ( R X g d µ ) = R ( µ ( X )) and N ( R X g d µ ) = N ( µ ( X )) .Proof. Consider an arbitrary u ∈ N ( R X g d µ ). In view of Remark B.1, then ν := u ∗ µu is abounded measure with R X g d ν = u ∗ ( R X g d µ ) u = 0 and ν ( { g ≤ } ) = 0. Thus, ν ( X ) = 0.Therefore, u ∗ µ ( X ) u = 0. Since, by virtue of µ ( X ) ∈ C q × q < and Remark A.31, we have µ ( X ) ∈ C q × q AD , we conclude then u ∈ N ( µ ( X )). So we have N ( R X g d µ ) ⊆ N ( µ ( X )). Observe that, dueto Lemma B.8, furthermore N ( µ ( X )) ⊆ N ( R X g d µ ) holds true. Hence, N ( R X g d µ ) = N ( µ ( X )).Using Remarks B.3 and A.10, we can then easily infer R ( R X g d µ ) = R ( µ ( X )).We will particularly apply the following result on integrable functions f satisfying Re f > f > Lemma B.10.
Let f ∈ L ( µ ) , let η, θ ∈ R , and let g := η Re f + θ Im f . Suppose that µ ( { g ≤ } ) = O q × q . Then g ∈ L ( µ ) with R (( R X f d µ ) ∗ ) = R ( R X f d µ ) , R (cid:18)Z X g d µ (cid:19) = R ( µ ( X )) , N (cid:18)Z X g d µ (cid:19) = N ( µ ( X )) , (B.1) R (cid:18)Z X f d µ (cid:19) = R ( µ ( X )) , and N (cid:18)Z X f d µ (cid:19) = N ( µ ( X )) . (B.2)90 roof. Because of Remarks B.3 and B.2, the real-valued function g belongs to L ( µ ). Hence,Lemma B.9 yields (B.1). Let h ∈ { f, f } . By virtue of Remark B.3, we have h ∈ L ( µ ).Consider an arbitrary u ∈ N ( R X h d µ ). In view of Remark B.1, then ν := u ∗ µu is a boundedmeasure and R X h d ν = u ∗ ( R X h d µ ) u = 0. Remarks B.3 and B.2 provide us R X Re h d ν = 0and R X Im h d ν = 0. Regarding Re h = Re f as well as Im h = Im f if h = f and Im h = − Im f if h = f , we thus obtain R X Re f d ν = 0 and R X Im f d ν = 0. Remark B.2 implies R X g d ν = 0. The assumption ν ( { g ≤ } ) = 0 yields then ν ( X ) = 0. So we have u ∗ µ ( X ) u =0. Since, by virtue of µ ( X ) ∈ C q × q < and Remark A.31, we have µ ( X ) ∈ C q × q AD , hence u ∈N ( µ ( X )). Observe that, due to Lemma B.8, furthermore N ( µ ( X )) ⊆ N ( R X h d µ ) holds true.Consequently, N ( R X h d µ ) = N ( µ ( X )). Using Remarks B.3 and A.10, we easily conclude h ∈ L ( µ ) and R ( R X h d µ ) = R (( R X h d µ ) ∗ ) = R ( µ ( X )). Choosing the appropriate h , we thusobtain (B.2) and R (( R X f d µ ) ∗ ) = R ( µ ( X )) = R ( R X f d µ ).We end this section with a matricial version of Lebesgue’s dominated convergence theorem : Proposition B.11 ( [26, Prop. A.6]) . Let f, f , f , . . . : X → C be measurable functions andlet g ∈ L ( µ ) be such that lim n →∞ f n ( x ) = f ( x ) for τ -almost all x ∈ X and | f n ( x ) | ≤ | g ( x ) | for all n ∈ N and τ -almost all x ∈ X hold true. Then the functions f, f , f , . . . belong to L ( µ ) and lim n →∞ R X f n d µ = R X f d µ . C. The Stieltjes transform of non-negative Hermitian measures
In this section, we consider a non-empty closed subset Ω of R and a non-negative Hermitian q × q measure σ on (Ω , B Ω ). So Ω is also a closed subset of C , whereas G := C \ Ω isan open subset of C . Observe that, for each t ∈ Ω, the function h t : G → C defined by h t ( z ) := 1 / ( t − z ) is holomorphic. Consider an arbitrary z ∈ G and let d z := inf x ∈ Ω | x − z | .Then d z > / | t − z | ≤ /d z for all t ∈ Ω. Consequently, the function g z : Ω → C defined by g z ( t ) := 1 / ( t − z ) belongs to L ( σ ). For each closed disk K ⊆ G , we have with d K := inf ( x,w ) ∈ Ω × K | x − w | , furthermore d K > / | t − z | ≤ /d K . By means of that, onecan check that the matrix-valued function ˆ S σ : G → C q × q defined byˆ S σ ( z ) := Z Ω t − z σ (d t ) (C.1)is holomorphic (see, e. g. [20, Satz 5.8, p. 147, Kapitel IV]). The mapping σ ˆ S σ is called Stieltjes transformation . Accordingly, the function ˆ S σ itself is called Stieltjes transform of σ .For Ω = R , the restriction of ˆ S σ onto Π + is exactly the R -Stieltjes transform S σ of σ introducedin Definition 5.1. Thus, we obtain: Lemma C.1.
Denote by F the restriction of ˆ S σ onto Π + . Then F ∈ R ,q (Π + ) and thespectral measure σ F of F fulfills σ F ( R \ Ω) = O q × q and σ F ( B ) = σ ( B ) for all B ∈ B Ω .Proof. Let χ : B R → C q × q be defined by χ ( B ) := σ ( B ∩ Ω). Then χ is a non-negative Hermitian q × q measure on ( R , B R ) satisfying F ( z ) = R R ( t − z ) − χ (d t ) for all z ∈ Π + . By virtue ofProposition 5.2, then F ∈ R ,q (Π + ) and χ = σ F follow.For Ω = [ α, β ] the function ˆ S σ coincides with the [ α, β ]-Stieltjes transform ¨ S σ of σ introducedin Definition 5.3. By virtue of (C.1), we easily see:91 emark C.2 . For all z ∈ C \ Ω, we haveRe ˆ S σ ( z ) = Z Ω t − Re z | t − z | σ (d t ) and Im ˆ S σ ( z ) = Z Ω Im z | t − z | σ (d t ) . Lemma C.3.
For all z ∈ C \ R , the matrix z Im ˆ S σ ( z ) is non-negative Hermitian. Fur-thermore, Im ˆ S σ ( x ) = O q × q for all x ∈ R \ Ω . Moreover, for all w ∈ C with Re w < inf Ω ,the matrix Re ˆ S σ ( w ) is non-negative Hermitian and, for all w ∈ C with Re w > sup Ω , thematrix − Re ˆ S σ ( w ) is non-negative Hermitian .Proof. Except for Im ˆ S σ ( x ) = O q × q for all x ∈ R \ Ω, the assertions are a consequence ofRemark C.2. Consider now an arbitrary x ∈ R \ Ω. The matrix-valued function ˆ S σ is holo-morphic. Hence, the two sequences ( ± Im ˆ S σ ( x ± i /n )) ∞ n =1 converge to ± Im ˆ S σ ( x ), resp. Asalready mentioned, we have z Im ˆ S σ ( z ) ∈ C q × q < for all z ∈ C \ R . In particular, the sequences( ± Im ˆ S σ ( x ± i /n )) ∞ n =1 both consist of non-negative Hermitian matrices. Consequently, we ob-tain ± Im ˆ S σ ( x ) ∈ C q × q < for their limits, implying Im ˆ S σ ( x ) = O q × q .Using Proposition B.11, one can easily deduce the following representations for σ (Ω) vialimits along the imaginary axis (cf. [26, Lem. A.8(c)]): Lemma C.4.
The following equations hold true: lim y →∞ y Re ˆ S σ (i y ) = O q × q , lim y →∞ y Im ˆ S σ (i y ) = σ (Ω) , and lim y →∞ i y ˆ S σ (i y ) = − σ (Ω) . The following lemma in particular shows that the matrix-valued function ˆ S σ has constantcolumn and null space on C \ [inf Ω , sup Ω]. Lemma C.5 (cf. [26, Lem. A.8(b)]) . For all z ∈ C \ [inf Ω , sup Ω] , the equations R ( ˆ S σ ( z )) = R ( σ (Ω)) and N ( ˆ S σ ( z )) = N ( σ (Ω)) (C.2) hold true. Furthermore, R (Im ˆ S σ ( z )) = R ( σ (Ω)) and N (Im ˆ S σ ( z )) = N ( σ (Ω)) (C.3) are valid for all z ∈ C \ R and R (Re ˆ S σ ( w )) = R ( σ (Ω)) and N (Re ˆ S σ ( w )) = N ( σ (Ω)) (C.4) are fulfilled for all w ∈ C with Re w < inf Ω or Re w > sup Ω .Proof. For each ζ ∈ C \ Ω, let g ζ : Ω → C be defined by g ζ ( t ) := 1 / ( t − ζ ). As alreadymentioned at the beginning of this section, we have g ζ ∈ L ( σ ) for each ζ ∈ C \ Ω. Because ofRemarks B.3 and B.2, then the functions Re g ζ and Im g ζ both belong to L ( σ ) and, in view ofˆ S σ ( ζ ) = R Ω g ζ d σ , fulfill Re ˆ S σ ( ζ ) = R Ω Re g ζ d σ and Im ˆ S σ ( ζ ) = R Ω Im g ζ d σ for each ζ ∈ C \ Ω.Furthermore, we have Re g ζ ( t ) = ( t − Re ζ ) / | t − ζ | and Im g ζ ( t ) = Im ζ/ | t − ζ | for each ζ ∈ C \ Ω. Consider now an arbitrary z ∈ C \ R . In the case Im z >
0, we have Im g z > f = g z , η = 0, and θ = 1 yields then (C.3) and(C.2). If Im z <
0, then − Im g z > θ = −
1. Now let w ∈ C with Re w < inf Ω or Re w > sup Ω. In the case Re w < inf Ω, wehave Re g w > f = g w , η = 1, and θ = 0 yieldsthen (C.4) and (C.2). If Re w > sup Ω, then − Re g w > η = −
1. 92ecall that a matrix A ∈ C q × q is called EP matrix if R ( A ∗ ) = R ( A ) and that A is saidto be almost definite if each x ∈ C q with x ∗ Ax = 0 necessarily fulfills Ax = O q × . Thecorresponding classes are denoted by C q × q EP and C q × q AD (cf. Definition 4.16). In view of (C.1),we obviously have [ ˆ S σ ( z )] ∗ = ˆ S σ ( z ) for all z ∈ C \ Ω. Consequently, from Lemma C.5 wecan infer R ([ ˆ S σ ( z )] ∗ ) = R ( ˆ S σ ( z )), i. e., ˆ S σ ( z ) ∈ C q × q EP for all z ∈ C \ [inf Ω , sup Ω]. RegardingRemark A.31, the values of ˆ S σ satisfy a stronger condition: Lemma C.6.
For all z ∈ C \ [inf Ω , sup Ω] , we have ˆ S σ ( z ) ∈ C q × q AD .Proof. First consider an arbitrary z ∈ C \ R . Let η := z . Lemma C.3 yields Im( η [ ˆ S σ ( z )]) ∈ C q × q < . From Lemma C.5 we can infer rank Im ˆ S σ ( z ) = rank ˆ S σ ( z ), implying rank Im( η [ ˆ S σ ( z )]) =rank( η [ ˆ S σ ( z )]). Thus, we can apply Lemma A.33 to obtain η [ ˆ S σ ( z )] ∈ C q × q AD . By virtue ofRemark A.29, then ˆ S σ ( z ) ∈ C q × q AD follows. Now consider an arbitrary x ∈ R \ [inf Ω , sup Ω].Assume x > sup Ω. From Lemma C.3 we can infer then − ˆ S σ ( x ) = − Re ˆ S σ ( x ) ∈ C q × q < . Becauseof Remark A.31, hence − ˆ S σ ( x ) ∈ C q × q AD . By virtue of Remark A.29, thus ˆ S σ ( x ) ∈ C q × q AD . If x < inf Ω, then Lemma C.3 yields ˆ S σ ( x ) = Re ˆ S σ ( x ) ∈ C q × q < , implying ˆ S σ ( x ) ∈ C q × q AD byRemark A.31. Lemma C.7.
For all z ∈ C \ R , the following matrix inequalities hold true: h ˆ S σ ( z ) i ∗ (cid:20) z Im ˆ S σ ( z ) (cid:21) † h ˆ S σ ( z ) i σ (Ω) and h ˆ S σ ( z ) i [ σ (Ω)] † h ˆ S σ ( z ) i ∗ z Im ˆ S σ ( z ) . Proof.
Consider an arbitrary z ∈ C \ R . Let g z : Ω → C be defined by g z ( t ) := 1 / ( t − z ) andlet Λ , Ξ : Ω → C q × q be defined by Λ( t ) := | g z ( t ) | I q and Ξ( t ) := I q . As already mentionedat the beginning of this section, we have g z ∈ L ( σ ). Because of Im g z / Im z = | g z | , we canconclude with Remarks B.2 and B.3 then | g z | ∈ L ( σ ). According to Remark B.7, hence thepair (Λ , Ξ) is left-integrable with respect to σ . Consequently, the matrix-valued function Λ σ ′ τ Ξ ∗ belongs to [ L ( τ )] q × q , where τ denotes the trace measure of σ and σ ′ τ is the trace derivativeof σ . Let Θ : Ω → C q × q be defined by Θ( t ) := g z ( t ) I q . Then Θ is measurable and fulfillsΛ σ ′ τ Ξ ∗ = Θ σ ′ τ Θ ∗ . Thus, the pair (Θ , Θ) is left-integrable with respect to σ , i. e., Θ ∈ L q × q ( σ ).By virtue of Remarks B.7 and C.2, we have Z Ω Θd σ Θ ∗ = Z Ω Θ σ ′ τ Θ ∗ d τ = Z Ω Λ σ ′ τ Ξ ∗ d τ = Z Ω Λd σ Ξ ∗ = Z Ω | g z | d σ = 1Im z Im ˆ S σ ( z ) . Furthermore, Ξ belongs to L q × q ( σ ) and fulfills Z Ω Ξd σ Ξ ∗ = Z Ω Ξ σ ′ τ Ξ ∗ d τ = Z Ω σ ′ τ d τ = σ (Ω) . In view of Remark B.4, then the pairs (Θ , Ξ) and (Ξ , Θ) are both left-integrable with respectto σ . Using Remarks B.7 and B.5, we then can infer Z Ω Θd σ Ξ ∗ = Z Ω g z d σ = ˆ S σ ( z ) and Z Ω Ξd σ Θ ∗ = (cid:18)Z Ω Θd σ Ξ ∗ (cid:19) ∗ = h ˆ S σ ( z ) i ∗ . The proof is completed by applying Lemma B.6 once with Φ = Θ and Ψ = Ξ and a secondtime with Φ = Ξ and Ψ = Θ. 93n combination with Lemma C.7, the following result reveals a certain minimality of thenon-negative Hermitian matrix σ (Ω) with respect to the Löwner partial order: Lemma C.8.
Let A ∈ C q × q < be such that R ( σ (Ω)) ⊆ R ( A ) and [ ˆ S σ ( z )] ∗ A † [ ˆ S σ ( z )] z Im ˆ S σ ( z ) for all z ∈ Π + . Then σ (Ω) A .Proof. Denote by F the restriction of ˆ S σ onto Π + . Because of Lemma C.1, then F ∈ R ,q (Π + )and σ F ( R ) = σ (Ω). Taking additionally into account Lemma C.5 and the assumptions, we get,for all z ∈ Π + , then R ( F ( z )) ⊆ R ( A ) and furthermore z Im F ( z ) − [ F ( z )] ∗ A † [ F ( z )] ∈ C q × q < .By virtue of Lemma A.26, hence the matrix h A F ( z )[ F ( z )] ∗ z Im F ( z ) i is non-negative Hermitian forall z ∈ Π + . The application of Lemma 4.5 thus yields σ F ( R ) A , implying the assertion. D. Particular pairs of matrices
Regular pairs of matrices considered in this section implement the extension of the set ofcomplex matrices by corresponding points at infinity analogous to the transition from the affineto the projective space. In this sense, they can be thought of as homogeneous coordinates.
Definition D.1.
An ordered pair [ P ; Q ] of complex p × q matrices P and Q is called p × q ma-trix pair . Such a pair is said to be regular if rank (cid:2) PQ (cid:3) = q and proper if rank Q = q .Each p × q matrix pair [ P ; Q ] generates a linear relation R := { ( Qv, P v ) : v ∈ C q } inthe C -vector space C p . In accordance with that, we associate to each p × q matrix pair[ P ; Q ] the linear subspaces R ( (cid:2) PQ (cid:3) ), R ( Q ), R ( P ), Q ( N ( P )), and P ( N ( Q )). Obviously, wehave Q ( N ( P )) ⊆ R ( Q ) and P ( N ( Q )) ⊆ R ( P ). Consequently, we get the inequalities0 ≤ dim R ( P ) − dim( P ( N ( Q ))) ≤ q . Lemma D.2.
Let [ P ; Q ] be a p × q matrix pair. Then dim R ( Q ) + dim( P ( N ( Q ))) = dim R " PQ = dim R ( P ) + dim( Q ( N ( P ))) . Proof.
Let U := N ( P ) and V := N ( Q ). The mappings φ : U → C p and ψ : V → C p definedby φ ( u ) := Qu and ψ ( v ) := P v , resp., are C -linear with ker φ = U ∩ V , φ ( U ) = Q ( U ), ker ψ = V ∩ U , and ψ ( V ) = P ( V ). Regarding dim ker φ +dim φ ( U ) = dim U and dim ker ψ +dim ψ ( V ) =dim V , then dim U = dim( U ∩ V ) + dim( Q ( N ( P ))) and dim V = dim( U ∩ V ) + dim( P ( N ( Q )))follow. The application of Remark A.3 yields dim R ( P )+dim U = q and dim R ( Q )+dim V = q .By virtue of Remark A.5, we see from Remark A.3 that dim R ( (cid:2) PQ (cid:3) ) + dim( U ∩ V ) = q . Takingall together, we obtaindim R ( Q ) + dim( P ( N ( Q ))) = q − dim( U ∩ V ) = dim R " PQ and dim R ( P ) + dim( Q ( N ( P ))) = q − dim( U ∩ V ) = dim R " PQ . A generalization of Remark A.3 for p × q matrix pairs immediately follows from Lemma D.2:94 emark D.3 . The equation dim R ( P ) − dim( P ( N ( Q ))) + dim( Q ( N ( P ))) = dim R ( Q ) holdstrue for each p × q matrix pair [ P ; Q ]. Lemma D.4.
Let [ P ; Q ] be a p × q matrix pair with dim R ( P ) − dim( P ( N ( Q ))) = q . Then rank P = q , rank Q = q , Q ( N ( P )) = { O p × } , and P ( N ( Q )) = { O p × } . In particular, thepair [ P ; Q ] is proper.Proof. By assumption, we have q = dim R ( P ) − dim( P ( N ( Q ))) ≤ dim R ( P ) = rank P ≤ q . Consequently, P ( N ( Q )) = { O p × } and rank P = q . Thus, N ( P ) = { O q × } and hence Q ( N ( P )) = { O p × } . We infer from P ( N ( Q )) = { O p × } furthermore N ( Q ) ⊆ N ( P ), implying N ( Q ) = { O q × } . Therefore, rank Q = q , i. e., [ P ; Q ] is proper.Each proper p × q matrix pair [ P ; Q ] is necessarily regular with P ( N ( Q )) = { O p × } anddim R ( P ) − dim( P ( N ( Q ))) = dim R ( P ). Furthermore, using Remarks A.3, A.4, and A.5, thefollowing result is readily checked: Remark
D.5 . Let [ P ; Q ] be a p × q matrix pair. Then [ P ; Q ] is proper if and only if det( Q ∗ Q ) =0. Furthermore, the following statements are equivalent:(i) [ P ; Q ] is regular.(ii) det( P ∗ P + Q ∗ Q ) = 0.(iii) N ( P ) ∩ N ( Q ) = { O q × } .Using Remark A.7, we obtain furthermore: Remark
D.6 . Let [ P ; Q ] be a p × q matrix pair and let V ∈ C q × s . Let φ := P V and let ψ := QV . Then [ φ ; ψ ] is a p × s matrix pair fulfilling ψ ∗ φ = V ∗ ( Q ∗ P ) V . If rank V = s and[ P ; Q ] is regular (resp., proper), then [ φ ; ψ ] is regular (resp., proper).It is readily checked that by the following definition an equivalence relation on the set of p × q matrix pairs is given: Definition D.7.
Two p × q matrix pairs [ P ; Q ] and [ S ; T ] are said to be equivalent if R ( (cid:2) PQ (cid:3) ) = R ( (cid:2) ST (cid:3) ). In this case, we write [ P ; Q ] ∼ = [ S ; T ]. Furthermore, denote by h [ P ; Q ] i thecorresponding equivalence class of a p × q matrix pair [ P ; Q ]. Remark
D.8 . Remarks A.23 and D.5 show that two regular p × q matrix pairs [ P ; Q ] and [ S ; T ]are equivalent if and only if there is an R ∈ C q × q with det R = 0 fulfilling S = P R and T = QR . Remark
D.9 . Each proper q × q matrix pair [ P ; Q ] satisfies det Q = 0 and [ P ; Q ] ∼ = [ P Q − ; I q ].Consequently, the set of equivalence classes of proper q × q matrix pairs can be identifiedwith the set of complex q × q matrices by means of h [ P ; Q ] i 7→ A := P Q − , where R ( Q ) = C q , R ( P ) = R ( A ), Q ( N ( P )) = N ( A ), and P ( N ( Q )) = { O q × } .In the remaining part of this section, we are concerned with reducing certain q × q ma-trix pairs [ P ; Q ], which satisfy a condition of the form R ( P ) ⊆ R ( M ) with a given complex q × p matrix M of rank r ≥
1, to r × r matrix pairs [ φ ; ψ ] without loosing any information: Lemma D.10.
Let θ ∈ C with | θ | = 1 and let [ P ; Q ] be a q × q matrix pair. Let A θ := Q + θP and let B θ := Q − θP . Suppose that det B θ = 0 and let K θ := A θ B − θ . Then R ( P ) = R ( I q − K θ ) , N ( P ) = B − θ ( N ( I q − K θ )) , (D.1) R ( Q ) = R ( I q + K θ ) , N ( Q ) = B − θ ( N ( I q + K θ )) , (D.2) Q ( N ( P )) = N ( I q − K θ ) , and P ( N ( Q )) = N ( I q + K θ ) . (D.3)95 urthermore, dim R ( P ) − dim( P ( N ( Q ))) = rank( I q − K θ ) + rank( I q + K θ ) − q and [ P ; Q ] isregular.Proof. We have K θ B θ = A θ and hence ( I q + K θ ) B θ = B θ + A θ = 2 Q and ( I q − K θ ) B θ = B θ − A θ = − θP . Thus, we can easily infer (D.2) and (D.1), using Remark A.8. From (D.1)and (D.2) we obtain Q ( N ( P )) = ( Q − θP )( N ( P )) = B θ B − θ ( N ( I q − K θ )) = N ( I q − K θ )and P ( N ( Q )) = ( Q − θP )( N ( Q )) = B θ B − θ ( N ( I q + K θ )) = N ( I q + K θ ) , i. e., (D.3). Due to Remark A.3, we havedim R ( I q + K θ ) + dim N ( I q + K θ ) = q, dim R ( I q − K θ ) + dim N ( I q − K θ ) = q. (D.4)Taking into account (D.1) and (D.3), we conclude from the first equation in (D.4) thendim R ( P ) − dim( P ( N ( Q ))) = dim R ( I q − K θ ) − dim N ( I q + K θ )= rank( I q − K θ ) + rank( I q + K θ ) − q. Lemma D.2 yields dim R ( (cid:2) PQ (cid:3) ) = dim R ( P ) + dim Q ( N ( P )). Because of (D.1), (D.3), and thesecond equation in (D.4), we infer rank (cid:2) PQ (cid:3) = q , i. e., [ P ; Q ] is regular.We think that the following result is well-known. However, we did not succeed in finding anavailable reference. Lemma D.11 (cf. [53, Lem. 1.6]) . Let [ P ; Q ] be a regular q × q matrix pair satisfying Im( Q ∗ P ) ∈ C q × q < . Let A := Q + i P and let B := Q − i P . Then det B = 0 and the ma-trix K := AB − satisfies k K k S ≤ .Proof. We have A ∗ A = ( Q ∗ − i P ∗ )( Q + i P ) = Q ∗ Q + i( Q ∗ P − P ∗ Q ) + P ∗ P = Q ∗ Q + P ∗ P − Q ∗ P )and B ∗ B = ( Q ∗ + i P ∗ )( Q − i P ) = Q ∗ Q − i( Q ∗ P − P ∗ Q ) + P ∗ P = Q ∗ Q + P ∗ P + 2 Im( Q ∗ P ) . In view of Remarks A.24 and A.25, in particular B ∗ B < Q ∗ Q + P ∗ P < O q × q follows. UsingRemark A.4 and Lemma A.27, we infer then N ( B ) = N ( B ∗ B ) ⊆ N ( Q ∗ Q + P ∗ P ). FromRemark D.5 we see furthermore det( P ∗ P + Q ∗ Q ) = 0. Consequently, N ( B ) = { O } , imply-ing det B = 0. Regarding KB = A , we have moreover B ∗ ( I q − K ∗ K ) B = B ∗ B − A ∗ A =4 Im( Q ∗ P ). Taking into account Remarks A.24 and A.25, we can conclude then I q − K ∗ K =4 B −∗ Im( Q ∗ P ) B − < O q × q . Thus, the application of Remark A.34 yields k K k S ≤ Lemma D.12.
Assume r ≤ q . Let U ∈ C q × r with U ∗ U = I r and let [ φ ; ψ ] be an r × r matrixpair. Let P := U φU ∗ and let Q := U ψU ∗ + P [ R ( U )] ⊥ . Then [ P ; Q ] is a q × q matrix pairwith R ( P ) ⊆ R ( U ) fulfilling det( P ∗ P + Q ∗ Q ) = det( φ ∗ φ + ψ ∗ ψ ) and Q ∗ P = U ( ψ ∗ φ ) U ∗ . Inparticular, [ P ; Q ] is regular if and only if [ φ ; ψ ] is regular. roof. Observe that R ( P ) = R ( U φU ∗ ) ⊆ R ( U ). Furthermore, we have P ∗ P = U φ ∗ U ∗ U φU ∗ = U φ ∗ φU ∗ . Let N := P [ R ( U )] ⊥ . By virtue of Remark A.11, then N ( N ) = R ( U ), implying N U = O . Consequently, we infer Q ∗ Q = U ψ ∗ U ∗ U ψU ∗ + U ψ ∗ ( N U ) ∗ + N U ψU ∗ + N ∗ N = U ψ ∗ ψU ∗ + N and Q ∗ P = U ψ ∗ U ∗ U φU ∗ + N ∗ U φU ∗ = U ψ ∗ φU ∗ + N U φU ∗ = U ψ ∗ φU ∗ . We are now going to show thatdet( P ∗ P + Q ∗ Q ) = det( φ ∗ φ + ψ ∗ ψ ) (D.5)holds true. Observe that, because of Remark D.5, the asserted equivalence immediately followsfrom (D.5). Using the already shown identities, we get P ∗ P + Q ∗ Q = U ( φ ∗ φ + ψ ∗ ψ ) U ∗ + N. (D.6)First assume r = q . Then the matrix U is unitary, implying N = O . Thus, (D.5) is aconsequence of (D.6).Now we consider the case r < q . Then there exists some V ∈ C q × ( q − r ) such that W := [ U, V ]is a unitary q × q matrix. In particular, we get W ∗ W = " U ∗ U U ∗ VV ∗ U V ∗ V = " I r O r × ( q − r ) O ( q − r ) × r I q − r and W W ∗ = U U ∗ + V V ∗ = I q . (D.7)Because of U ∗ U = I r and Remark A.12, we have U U ∗ = P R ( U ) . In view of Remark A.11, weobtain from the last equation in (D.7) thus V V ∗ = I q − U U ∗ = N . Taking additionally intoaccount (D.6) and (D.7), we hence can infer W ∗ ( P ∗ P + Q ∗ Q ) W = " U ∗ V ∗ [ U ( φ ∗ φ + ψ ∗ ψ ) U ∗ + V V ∗ ][ U, V ] = φ ∗ φ + ψ ∗ ψ O r × ( q − r ) O ( q − r ) × r I q − r . In particular, (D.5) holds true.
Proposition D.13 (cf. [3, Lem. 4.3]) . Let M ∈ C q × p with rank r ≥ , let u , u , . . . , u r be anorthonormal basis of R ( M ) , and let U := [ u , u , . . . , u r ] . Let [ P ; Q ] be a regular q × q matrixpair fulfilling Im( Q ∗ P ) ∈ C q × q < and R ( P ) ⊆ R ( M ) and let B := Q − i P . Then det B = 0 .Let φ := U ∗ P B − U and let ψ := U ∗ QB − U . Then [ φ ; ψ ] is a regular r × r matrix pairsatisfying ψ ∗ φ = ( B − U ) ∗ ( Q ∗ P )( B − U ) . Let S := U φU ∗ and let T := U ψU ∗ + P [ R ( M )] ⊥ .Then [ S ; T ] is a regular q × q matrix pair satisfying det( S ∗ S + T ∗ T ) = det( φ ∗ φ + ψ ∗ ψ ) and T ∗ S = B −∗ ( Q ∗ P ) B − . Furthermore, [ P ; Q ] ∼ = [ S ; T ] with S = P B − and T = QB − .Proof. We only consider here the case r < q . Then there exists some V ∈ C q × ( q − r ) such that W := [ U, V ] is a unitary q × q matrix. In particular, we get (D.7). Let A := Q + i P . FromLemma D.11 we see det B = 0 and that K := AB − satisfies k K k S ≤
1. Consequently thematrix L := W ∗ KW then satisfies k L k S ≤ L = " U ∗ V ∗ K [ U, V ] = " U ∗ KU U ∗ KVV ∗ KU V ∗ KV . B − A = − P and B + A = 2 Q hold true. Consequently, we obtaini2 ( I q − K ) = P B − and 12 ( I q + K ) = QB − . (D.8)Remark A.12 yields U U ∗ = P R ( M ) . Hence, U U ∗ P = P follows. With (D.8) and (D.7), we canthus conclude V ∗ ( I q − K ) = − V ∗ P B − = − V ∗ U U ∗ P B − = O ( q − r ) × q , implying V ∗ K = V ∗ . Regarding (D.7), we infer for the lower blocks of L then V ∗ KU = V ∗ U = O ( q − r ) × r and V ∗ KV = V ∗ V = I q − r . In particular, the lower right block V ∗ KV of L is unitary.Consequently, the application of Remark A.35 to L yields the block representation L = " U ∗ KU O r × ( q − r ) O ( q − r ) × r I q − r . (D.9)First we verify the assertions for the pair [ S ; T ]: Using (D.7) and (D.9), we obtain I q = U U ∗ + V V ∗ = U U ∗ U U ∗ + V V ∗ , K = W LW ∗ = [ U, V ] L " U ∗ V ∗ = U U ∗ KU U ∗ + V V ∗ and, consequently, I q − K = U U ∗ ( I q − K ) U U ∗ as well as I q + K = U U ∗ ( I q + K ) U U ∗ + 2 V V ∗ .Because of (D.8), then P B − = U U ∗ P B − U U ∗ = U φU ∗ and QB − = U U ∗ QB − U U ∗ + V V ∗ = U ψU ∗ + V V ∗ follow. In view of (D.7) and U U ∗ = P R ( M ) , we infer from Remark A.11 furthermore V V ∗ = I q − U U ∗ = P [ R ( M )] ⊥ . Thus, we get P B − = S and QB − = T . By virtue of Remark D.6, hence[ S ; T ] is a regular q × q matrix pair fulfilling T ∗ S = B −∗ ( Q ∗ P ) B − . In addition, Remark D.8yields [ P ; Q ] ∼ = [ S ; T ]. It remains to show the assertions involving the pair [ φ ; ψ ]: Regarding U ∗ U = I r and R ( U ) = R ( M ), we can apply Lemma D.12 to the r × r matrix pair [ φ ; ψ ] toobtain det( S ∗ S + T ∗ T ) = det( φ ∗ φ + ψ ∗ ψ ) and to see that [ φ ; ψ ] is regular and that T ∗ S = U ( ψ ∗ φ ) U ∗ holds true. From the last equation we can infer then ψ ∗ φ = U ∗ U ψ ∗ φU ∗ U = U ∗ T ∗ SU = U ∗ ( QB − ) ∗ ( P B − ) U = ( B − U ) ∗ ( Q ∗ P )( B − U ) . E. Linear fractional transformations of matrices
In this appendix, we consider a matricial generalization of the transformation z az + bcz + d of theextended complex plane. We thereby follow [13, Sec. 1.6], while restricting ourselves to theversion Z ( AZ + B )( CZ + D ) − with denominator on the right side. Let (cid:2) A BC D (cid:3) be the blockrepresentation of a complex ( p + q ) × ( p + q ) matrix M with p × p block A . If the set Q C,D := (cid:8) Z ∈ C p × q : det( CZ + D ) = 0 (cid:9) (resp., PQ C,D := (cid:8) ( P, Q ) ∈ C p × q × C q × q : det( CP + DQ ) = 0 (cid:9) )is non-empty, then let the linear fractional transformation Φ ( p,q ) M : Q C,D → C p × q (resp.,Ψ ( p,q ) M : PQ C,D → C p × q ) be defined byΦ ( p,q ) M ( Z ) := ( AZ + B )( CZ + D ) − (resp., Ψ ( p,q ) M ([ P ; Q ]) := ( AP + BQ )( CP + DQ ) − ) .
98n this context, the block matrix M = (cid:2) A BC D (cid:3) is called the generating matrix of the linearfractional transformation. For each matrix Z ∈ Q C,D , we obviously have (
Z, I q ) ∈ PQ C,D andΦ ( p,q ) M ( Z ) = Ψ ( p,q ) M (( Z, I q )). We first characterize the case, that the corresponding domain isnon-empty: Lemma E.1 ( [31, Lem. D.2]) . The following statements are equivalent:(i) Q C,D = ∅ .(ii) PQ C,D = ∅ .(iii) rank[ C, D ] = q . The composition of two linear fractional transformations is again a linear fractional trans-formation with generating matrix M emerging from ordinary matrix multiplication: Proposition E.2 (cf. [31, Propositions D.3 and D.4]) . Let (cid:2) A B C D (cid:3) and (cid:2) A B C D (cid:3) be the blockrepresentations of two given complex ( p + q ) × ( p + q ) matrices M and M with p × p block A and A , resp. Let (cid:2) A BC D (cid:3) be the block representation of the product M := M M with p × p block A .(a) Suppose that the set Q := { Z ∈ Q C ,D : Φ ( p,q ) M ( Z ) ∈ Q C ,D } is non-empty. Then Q ⊆ Q
C,D and Φ ( p,q ) M ( Z ) = Φ ( p,q ) M (Φ ( p,q ) M ( Z )) for all Z ∈ Q .(b) Suppose that the set PQ := { [ P ; Q ] ∈ PQ C ,D : Ψ ( p,q ) M ([ P ; Q ]) ∈ Q C ,D } is non-empty.Then PQ ⊆ PQ
C,D and Ψ ( p,q ) M ([ P ; Q ]) = Φ ( p,q ) M (Ψ ( p,q ) M ([ P ; Q ])) for all [ P ; Q ] ∈ PQ . In connection with the particular embedding of r × r matrix pairs into the class of q × q ma-trix pairs for r ≤ q considered in Lemma D.12, the following auxiliary result is of interest: Lemma E.3.
Suppose q ≥ and let r ∈ Z ,q − . Let [ U, V ] be the block representation ofa unitary q × q matrix W with q × r block U . Let (cid:2) A BC D (cid:3) be the block representation of acomplex q × q matrix M with q × q block A and let N := (cid:2) AW BWCW DW (cid:3) .(a) Let f ∈ C r × r and let F := f ⊕ O ( q − r ) × ( q − r ) . Then U f U ∗ ∈ Q C,D if and only if F ∈ Q CW,DW . In this case, Φ ( q,q ) M ( U f U ∗ ) = Φ ( q,q ) N ( F ) .(b) Let f, g ∈ C r × r , let F := f ⊕ O ( q − r ) × ( q − r ) , and let G := g ⊕ I q − r . Then ( U f U ∗ , U gU ∗ + P [ R ( U ) ⊥ ] ) ∈ PQ C,D if and only if ( F, G ) ∈ PQ CW,DW . In this case, Ψ ( q,q ) M (( U f U ∗ , U gU ∗ + P [ R ( U ) ⊥ ] )) = Ψ ( q,q ) N ([ F ; G ]) .Proof. (a) Because of W F W ∗ = U f U ∗ and W − = W ∗ , we have [( AW ) F + ( BW )] W − = A ( W F W ∗ ) + B = A ( U f U ∗ ) + B and similarly [( CW ) F + ( DW )] W − = C ( U f U ∗ ) + D .Consequently, (a) follows.(b) As in the proof of Lemma D.12, we have (D.7) and we can conclude V V ∗ = P [ R ( U )] ⊥ .Beside W F W ∗ = U f U ∗ , we get W GW ∗ = U gU ∗ + V V ∗ = U gU ∗ + P [ R ( U )] ⊥ . The equation[( AW ) F + ( BW ) G ] W − = A ( W F W ∗ ) + B ( W GW ∗ ) = A ( U f U ∗ ) + B ( U gU ∗ + P [ R ( U )] ⊥ ) thenfollows from W − = W ∗ . Similarly, we obtain moreover [( CW ) F +( DW ) G ] W − = C ( U f U ∗ )+ D ( U gU ∗ + P [ R ( U )] ⊥ ). Consequently, (b) follows.99 . Holomorphic matrix-valued functions Let G be a domain , i. e., an open, non-empty, and connected subset of C . A matrix-valuedfunction F : G → C p × q is said to be holomorphic if all entries f jk : G → C of F = [ f jk ] j =1 ,...,pk =1 ,...,q are holomorphic functions. In this case, the matrix-valued function F admits, for each z ∈ G ,a unique power series representation F ( z ) = P ∞ n =0 ( z − z ) n A n . The corresponding disk ofconvergence coincides with the largest open disk with center z lying entirely in G . Thecoefficients A n = [ a jk,n ] j =1 ,...,pk =1 ,...,q are given by the Taylor series f jk ( z ) = P ∞ n =0 a jk,n ( z − z ) n at z . Setting F ( n ) with the n -th derivatives f ( n ) jk of the infinitely differentiable functions f jk , wehave A n = n ! F ( n ) ( z ) . Basic results on holomorphic functions can be generalized to the matrixcase considered here in an appropriate way:
Remark
F.1 . Let F : G → C p × q be holomorphic, let U ∈ C r × p , and let V ∈ C q × s . Then H := U F V is holomorphic with H ( n ) = U F ( n ) V for all n ∈ N .The Cauchy product for sequences of matrices determines the coefficients of the product oftwo matrix-valued power series: Remark
F.2 . Let F : G → C p × q and G : G → C q × r be two holomorphic functions. Let z ∈ G and let the sequences ( A n ) ∞ n =0 and ( B n ) ∞ n =0 be given by A n := n ! F ( n ) ( z ) and B n := n ! G ( n ) ( z ),resp. Then H := F G is holomorphic and the sequence ( C n ) ∞ n =0 given by C n := n ! H ( n ) ( z )coincides with the Cauchy product of ( A n ) ∞ n =0 and ( B n ) ∞ n =0 .If, in the case p = q , the values F ( z ) of the holomorphic matrix-valued function F areinvertible matrices for all z ∈ G , then the function G : G → C q × q defined by G ( z ) := [ F ( z )] − isholomorphic as well. Now suppose that F satisfies only the weaker condition of having constantcolumn space R ( F ( z )) and constant null space N ( F ( z )), independent of the argument z ∈ G .Then, even in the case p = q , the function G : G → C q × p defined by G ( z ) := [ F ( z )] † turnsout to be holomorphic. Furthermore, the sequences of Taylor coefficients of G and F bothbelong to the class introduced in Notation F.3 below and are mutually reciprocal in the senseof Definition 3.49: Notation
F.3 . Let D p × q,κ be the set of all sequences ( s j ) κj =0 of complex p × q matrices satisfying S κj =0 R ( s j ) ⊆ R ( s ) and N ( s ) ⊆ T κj =0 N ( s j ).The following is a specification of a result due to Campbell and Meyer [4, Thm. 10.5.4]: Proposition F.4.
Let F : G → C p × q be holomorphic. Then the following statements areequivalent:(i) The function G : G → C q × p defined by G ( z ) := [ F ( z )] † is holomorphic.(ii) R ( F ( z )) = R ( F ( w )) and N ( F ( z )) = N ( F ( w )) for all z, w ∈ G .(iii) ( n ! F ( n ) ( z )) ∞ n =0 ∈ D p × q, ∞ for all z ∈ G .If (i) is fulfilled and z ∈ G , then ( n ! G ( n ) ( z )) ∞ n =0 is exactly the reciprocal sequence associatedto ( n ! F ( n ) ( z )) ∞ n =0 .Proof. The equivalence of (i) and (ii) is an immediate consequence of [33, Prop. 8.4]. Let (i)be fulfilled. Consider an arbitrary z ∈ G . Because of [33, Thm. 8.9 and 4.21], the sequence100 n ! F ( n ) ( z )) ∞ n =0 belongs to D p × q, ∞ and ( n ! G ( n ) ( z )) ∞ n =0 is exactly the reciprocal sequence as-sociated to ( n ! F ( n ) ( z )) ∞ n =0 . In particular, (iii) holds true. Conversely, suppose that (iii) isfulfilled. From [33, Thm. 8.9] we can then infer that the function G is holomorphic in all points z ∈ G . Consequently, (i) holds true.Next, we give analogous results for power series expansions at z = ∞ . To that end, let ρ ∈ (0 , ∞ ) and suppose that the improper open annulus C ρ := { z ∈ C : | z | > ρ } is entirelycontained in G . Furthermore, let a holomorphic matrix-valued function F : G → C p × q be given,admitting the series representation F ( z ) = ∞ X n =0 z − n C n (F.1)for all z ∈ C ρ with certain complex p × q matrices C , C , C , . . . This is the matricial version ofa special case of the general situation of a given complex-valued function f which is holomorphicin an annulus A := { z ∈ C : r < | z − c | < R } centered at c ∈ C with radii 0 ≤ r < R ≤ ∞ . Asis well known, such a function f has a Laurent series f ( z ) = P ∞ ℓ = −∞ a ℓ ( z − c ) ℓ at the point c converging on A with uniquely determined coefficients a ℓ ∈ C . In the particular situation ofinterest considered here, we have c = 0, R = ∞ , and a ℓ = 0 for all ℓ ∈ N . This case can beeasily reduced to the ordinary power series expansion of holomorphic functions, discussed atthe beginning of this section. By means of the substitution z w := 1 /z , we can proceed toa holomorphic function Φ defined on the open disk B /ρ := { w ∈ C : | w | < /ρ } with Taylorseries Φ( w ) = P ∞ n =0 w n C n at the point w = 0: Lemma F.5.
Let F : G → C p × q be holomorphic, admitting for all z ∈ C ρ the series repre-sentation (F.1) with certain complex p × q matrices C , C , C , . . . Then lim ζ → F (1 /ζ ) = C and the matrix-valued function Φ : B /ρ → C p × q defined by Φ( w ) := F (1 /w ) for w = 0 andby Φ(0) := lim ζ → F (1 /ζ ) is holomorphic with n ! Φ ( n ) (0) = C n for all n ∈ N . We continue with the analogue of Remark F.1 for power series expansion at z = ∞ : Remark
F.6 . Let F : G → C p × q be holomorphic, admitting the series representation (F.1) forall z ∈ C ρ with certain complex p × q matrices C , C , C , . . . Let U ∈ C r × p and let V ∈ C q × s .Then H := U F V is holomorphic and H ( z ) = P ∞ n =0 z − n ( U C n V ) for all z ∈ C ρ .Likewise, Remark F.2 can be modified in a well-known matter for power series expansion at z = ∞ : Lemma F.7.
Let F : G → C p × q and G : G → C q × r be holomorphic functions, admitting theseries representations F ( z ) = P ∞ n =0 z − n C n and G ( z ) = P ∞ n =0 z − n D n for all z ∈ C ρ with certaincomplex p × q matrices C , C , C , . . . and certain complex q × r matrices D , D , D , . . . , resp.Let H := F G and denote by ( E n ) ∞ n =0 the Cauchy product of ( C n ) ∞ n =0 and ( D n ) ∞ n =0 . Then H is holomorphic and H ( z ) = P ∞ n =0 z − n E n for all z ∈ C ρ . Using Proposition F.4, we are able to expand the function z [ F ( z )] † under certain con-ditions at z = ∞ into a series with coefficients given, according to Definition 3.49, by thereciprocal sequence associated to the sequence ( C n ) ∞ n =0 from (F.1): Lemma F.8.
Let F : G → C p × q be holomorphic and let ( C n ) ∞ n =0 be a sequence of complex p × q matrices such that (F.1) and furthermore R ( F ( z )) = R ( C ) and N ( F ( z )) = N ( C ) hold rue for all z ∈ C ρ . Let G : C ρ → C q × p be defined by G ( z ) := [ F ( z )] † and denote by ( D n ) ∞ n =0 thereciprocal sequence associated to ( C n ) ∞ n =0 . Then G is holomorphic and G ( z ) = P ∞ n =0 z − n D n for all z ∈ C ρ .Proof. According to Lemma F.5, we proceed to a holomorphic function Φ : B /ρ → C p × q , whichsatisfies n ! Φ ( n ) (0) = C n for all n ∈ N . Consider an arbitrary w ∈ B /ρ . If w = 0, then Φ( w ) = C . In the case w = 0, we see that z := 1 /w belongs to C ρ and that Φ( w ) = F ( z ). Consequently, R (Φ( w )) = R ( C ) and N (Φ( w )) = N ( C ) for all w ∈ B /ρ . In particular, R (Φ( w )) and N (Φ( w )) are independent of w ∈ B /ρ . Let Ψ : B /ρ → C q × p be defined by Ψ( w ) := [Φ( w )] † .From Proposition F.4 we see then that Ψ is holomorphic and that ( n ! Ψ ( n ) (0)) ∞ n =0 is exactlythe reciprocal sequence associated to ( n ! Φ ( n ) (0)) ∞ n =0 . Hence, we have n ! Ψ ( n ) (0) = D n for all n ∈ N and thus Ψ( w ) = P ∞ n =0 w n D n for all w ∈ B /ρ . Consider an arbitrary z ∈ C ρ . Then w := 1 /z belongs to B /ρ \ { } and we have Ψ( w ) = [ F (1 /w )] † , implying G ( z ) = [ F ( z )] † = Ψ( w ) = ∞ X n =0 w n D n = ∞ X n =0 z − n D n . In the remaining part of this section, let G be again an arbitrary domain. Next, we considerthe matricial generalization of a special class of holomorphic functions, which is well studied,especially in the generic case of G being the open unit disk: Notation
F.9 . Denote by S p × q ( G ) the set of all functions S : G → C p × q , which are holomorphicin G and satisfy k S ( z ) k S ≤ z ∈ G .The matrix-valued functions belonging to S p × q ( G ) are called Schur functions (in G ) . Lemma F.10 (cf. [30, Lem. 3.9]) . Let S ∈ S p × q ( G ) and let U, V ∈ C p × q with U U ∗ = I p and V ∗ V = I q . For all z, w ∈ G , then R ( U + S ( z )) = R ( U + S ( w )) and N ( V + S ( z )) = N ( V + S ( w )) . We end this section with some remarks concerning meromorphic matrix-valued functions. Asubset D of G is said to be discrete in G if G does not contain any accumulation point of D .So, according to the identity theorem for holomorphic functions, two holomorphic functions F, G : G → C p × q coincide if and only if the set { z ∈ G : F ( z ) = G ( z ) } is not discrete in G .Speaking in the following of a discrete subset of G , we always mean a subset of G , which isdiscrete in G . For such a discrete subset D of G , the set G \ D is a domain.A complex-valued function f is said to be meromorphic in G if there exists a discrete subset P ( f ) of G such that f is a holomorphic function defined on the domain G \ P ( f ), which ineach point from P ( f ) has a pole (of positive order). In particular, each holomorphic function f : G → C is meromorphic in G with P ( f ) = ∅ . We call a C p × q -valued function F meromorphicin G if all entries f jk of F = [ f jk ] j =1 ,...,pk =1 ,...,q are complex-valued functions meromorphic in G .In this case, the union P ( F ) := S pj =1 S qk =1 P ( f jk ) of the sets of poles of all entries f jk is adiscrete subset of G . In particular, each holomorphic function F : G → C p × q is meromorphicin G with P ( F ) = ∅ . Since G is assumed to be connected, the set of complex-valued functionsmeromorphic in G has the algebraic structure of a field. Using the arithmetic of this field, theusual operations from matrix algebra can be formally carried over to matrix-valued functions,which are meromorphic in G . Thus, corresponding sums and products of such matrix-valuedfunctions are again meromorphic in G . Furthermore, it is readily checked that the determinant102et F of a C q × q -valued function F meromorphic in G is a complex-valued function, whichis meromorphic in G . If det F does not identically vanish, then the mapping F − given byformal matrix inversion of F (seen as a matrix with entries in the field of complex-valuedfunctions meromorphic in G ) is again a C q × q -valued function, which is meromorphic in G withnot identically vanishing determinant. References [1] Akhiezer, N.I.:
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