Minimal Rank Decoupling of Full-Lattice CMV Operators with Scalar- and Matrix-Valued Verblunsky Coefficients
aa r X i v : . [ m a t h . SP ] F e b MINIMAL RANK DECOUPLING OF FULL-LATTICE CMV OPERATORSWITH SCALAR- AND MATRIX-VALUEDVERBLUNSKY COEFFICIENTS
STEPHEN CLARK, FRITZ GESZTESY, AND MAXIM ZINCHENKO
Abstract.
Relations between half- and full-lattice CMV operators with scalar- and matrix-valuedVerblunsky coefficients are investigated. In particular, the decoupling of full-lattice CMV oper-ators into a direct sum of two half-lattice CMV operators by a perturbation of minimal rank isstudied. Contrary to the Jacobi case, decoupling a full-lattice CMV matrix by changing one ofthe Verblunsky coefficients results in a perturbation of twice the minimal rank. The explicit formfor the minimal rank perturbation and the resulting two half-lattice CMV matrices are obtained.In addition, formulas relating the Weyl–Titchmarsh m -functions (resp., matrices) associated withthe involved CMV operators and their Green’s functions (resp., matrices) are derived. Introduction
CMV operators are a special class of unitary semi-infinite or doubly-infinite five-diagonal matriceswhich received enormous attention in recent years. We refer to (2.8) and (3.18) for the explicit formof doubly infinite CMV operators on Z in the case of scalar, respectively, matrix-valued Verblunskycoefficients. For the corresponding half-lattice CMV operators we refer to (2.16) and (3.26).The actual history of CMV operators (with scalar Verblunsky coefficients) is somewhat intrigu-ing: The corresponding unitary semi-infinite five-diagonal matrices were first introduced in 1991 byBunse–Gerstner and Elsner [15], and subsequently discussed in detail by Watkins [82] in 1993 (cf. thediscussion in Simon [73]). They were subsequently rediscovered by Cantero, Moral, and Vel´azquez(CMV) in [17]. In [71, Sects. 4.5, 10.5], Simon introduced the corresponding notion of unitary doublyinfinite five-diagonal matrices and coined the term “extended” CMV matrices. For simplicity, wewill just speak of CMV operators, irrespective of whether or not they are half-lattice or full-latticeoperators. We also note that in a context different from orthogonal polynomials on the unit circle,Bourget, Howland, and Joye [14] introduced a family of doubly infinite matrices with three setsof parameters which, for special choices of the parameters, reduces to two-sided CMV matrices on Z . Moreover, it is possible to connect unitary block Jacobi matrices to the trigonometric momentproblem (and hence to CMV matrices) as discussed by Berezansky and Dudkin [11], [12].The relevance of this unitary operator U on ℓ ( Z ) m , more precisely, the relevance of the corre-sponding half-lattice CMV operator U + , in ℓ ( N ) m is derived from its intimate relationship withthe trigonometric moment problem and hence with finite measures on the unit circle ∂ D . (Here N = N ∪ { } .) Following [19], [20], [49], [50], and [87], this will be reviewed in some detail, and alsoextended in certain respects, in Sections 2 and 3, as this material is of fundamental importance tothe principal topics (such as decoupling of full-lattice CMV operators into direct sums of left and Mathematics Subject Classification.
Primary 34E05, 34B20, 34L40, 34A55.
Key words and phrases.
CMV operators, orthogonal polynomials, finite difference operators, Weyl–Titchmarshtheory, finite rank perturbations.Appeared in
Difference Equations and Applications , Proceeding of the 14th International Conference on DifferenceEquations and Applications, Istanbul, July 21–25, 2008, M. Bohner, Z. Doˇsl´a, G. Ladas, M. ¨Unal, and A. Zafer (eds.),U˘gur–Bah¸ce¸sehir University Publishing Company, Istanbul, Turkey, 2009, pp. 19–59. right half-lattice CMV operators and a similar result for associated Green’s functions) discussed inthis paper, but we also refer to the monumental two-volume treatise by Simon [71] (see also [70] and[72]) and the exhaustive bibliography therein. For classical results on orthogonal polynomials on theunit circle we refer, for instance, to [6], [41]–[43], [53], [75]–[77], [80], [81]. More recent referencesrelevant to the spectral theoretic content of this paper are [22], [38]–[40], [49], [50], [51], [66], [69], and[87]. The full-lattice CMV operators U on Z are closely related to an important, and only recentlyintensively studied, completely integrable nonabelian version of the defocusing nonlinear Schr¨odingerequation (continuous in time but discrete in space), a special case of the Ablowitz–Ladik system.Relevant references in this context are, for instance, [1]–[5], [37], [44]–[47], [57], [59]–[62], [68], [79],and the literature cited therein. We emphasize that the case of matrix-valued coefficients α k isconsiderably less studied than the case of scalar coefficients.We should also emphasize that while there is an extensive literature on orthogonal matrix-valuedpolynomials on the real line and on the unit circle, we refer, for instance, to [7], [9], [10, Ch. VII],[13], [16], [18], [23]–[34], [35], [36], [54], [55], [56], [58], [63]–[65], [67], [74], [83]–[86], and the largebody of literature therein, the case of CMV operators with matrix-valued Verblunsky coefficientsappears to be a much less explored frontier. The only references we are aware of in this context areSimon’s treatise [71, Part 1, Sect. 2.13] and the recent papers [8], [19], [21], and [73].Finally, a brief description of the content of each section in this paper: In Section 2 we review,and in part, extend the basic Weyl–Titchmarsh theory for half-lattice CMV operators with scalarVerblunsky coefficients originally derived in [49], and recall its intimate connections with transfermatrices and orthogonal Laurent polynomials. The principal result of this section, Theorem 2.3, thenprovides a necessary and sufficient condition for the difference between the full-lattice CMV operator U and its “decoupling” into a direct sum of appropriate left and right half-lattice CMV operators tobe of rank one. The same result is also derived for the resolvent differences of U and the resolvent ofits decoupling into a direct sum of left and right half-lattice CMV operators. Theorem 2.3 is in sharpcontrast to the familiar Jacobi case, since decoupling a full-lattice CMV matrix by changing one ofthe Verblunsky coefficients results in a perturbation of rank two. While this difference compared toJacobi operators was noticed first by Simon [71, Sect. 4.5], we explore it further here and providea complete discussion of this decoupling phenomenon, including its extension to the matrix-valuedcase, which represents a new result. We conclude this section with a discussion of half-lattice Green’sfunctions in Lemma 2.7, extending a result in [49].In Section 3 we develop all these results for CMV operators with m × m , m ∈ N , matrix-valuedVerblunsky coefficients. In particular, in Theorem 3.6, the principal result of this section, we providea necessary and sufficient condition for the difference between the full-lattice CMV operator U andits decoupling into a direct sum of appropriate left and right half-lattice CMV operators to be ofminimal rank m .Finally, Appendix A summarizes basic facts on matrix-valued Caratheodory and Schur functionsrelevant to this paper. 2. CMV operators with scalar coefficients
This section is devoted to a study of CMV operators associated with scalar Verblunsky coefficients.We derive a criterion under which a difference of a full-lattice CMV operator and a direct sum of twohalf-lattice CMV operators is of rank one. The same condition will also imply a similar result forthe resolvents of these operators. At the end of the section we establish relations that hold betweenWeyl–Titchmarsh m -functions associated with the above operators and derive explicit expressionsfor half-lattice Green’s matrices.We start by introducing basic notations used throughout this paper. Let s( Z ) be the space ofcomplex-valued sequences and ℓ ( Z ) ⊂ s( Z ) be the usual Hilbert space of all square summable INIMAL RANK DECOUPLING OF CMV OPERATORS 3 complex-valued sequences with scalar product ( · , · ) ℓ ( Z ) linear in the second argument. The standardbasis in ℓ ( Z ) is denoted by { δ k } k ∈ Z , δ k = ( . . . , , . . . , , |{z} k , , . . . , , . . . ) ⊤ , k ∈ Z . (2.1)For m ∈ N and J ⊆ R an interval, we will identify ⊕ mj =1 ℓ ( J ∩ Z ) and ℓ ( J ∩ Z ) ⊗ C m andthen use the simplified notation ℓ ( J ∩ Z ) m . For simplicity, the identity operators on ℓ ( J ∩ Z ) and ℓ ( J ∩ Z ) m are abbreviated by I without separately indicating its dependence on m and J . Theidentity m × m matrix is denoted by I m .Throughout this section we make the following basic assumption: Hypothesis 2.1.
Let α = { α k } k ∈ Z ∈ s( Z ) be a sequence of complex numbers such that α k ∈ D , k ∈ Z . (2.2)Given a sequence α satisfying (2.2), we define the following sequence of positive real numbers { ρ k } k ∈ Z by ρ k = (cid:2) − | α k | (cid:3) / , k ∈ Z . (2.3)Following Simon [71], we call { α k } k ∈ Z the Verblunsky coefficients in honor of Verblunsky’s pio-neering work in the theory of orthogonal polynomials on the unit circle [80], [81].Next, we also introduce a sequence of 2 × k byΘ k = (cid:18) − α k ρ k ρ k α k (cid:19) , k ∈ Z , (2.4)and two unitary operators V and W on ℓ ( Z ) by their matrix representations in the standard basisof ℓ ( Z ) as follows, V = . . . Θ k − Θ k . . . , W = . . . Θ k − Θ k +1 . . . , (2.5)where (cid:18) V k − , k − V k − , k V k, k − V k, k (cid:19) = Θ k , (cid:18) W k, k W k, k +1 W k +1 , k W k +1 , k +1 (cid:19) = Θ k +1 , k ∈ Z . (2.6)Moreover, we introduce the unitary operator U on ℓ ( Z ) by U = V W, (2.7)
S. CLARK, F. GESZTESY, AND M. ZINCHENKO or in matrix form, in the standard basis of ℓ ( Z ), by U = . . . . . . . . . . . . . . . − α ρ − − α − α − α ρ ρ ρ ρ − ρ α − ρ − α α α ρ − α ρ − α α − α ρ ρ ρ ρ ρ α ρ − α α α ρ
0. . . . . . . . . . . . . . . (2.8)= ρ − ρ δ even S −− + ( α − ρ δ even − α + ρ δ odd ) S − − αα + + ( αρ + δ even − α ++ ρ + δ odd ) S + + ρ + ρ ++ δ odd S ++ , (2.9)where δ even and δ odd denote the characteristic functions of the even and odd integers, δ even = χ Z , δ odd = 1 − δ even = χ Z +1 (2.10)and S ± , S ++ , S −− denote the shift operators acting upon s( Z ), that is, S ± f ( · ) = f ± ( · ) = f ( · ± f ∈ s( Z ), S ++ = S + S + , and S −− = S − S − . Here the diagonal entries in the infinite matrix (2.8)are given by U k,k = − α k α k +1 , k ∈ Z .As explained in the introduction, in the recent literature on orthogonal polynomials on the unitcircle, such operators U are frequently called CMV operators.Next we recall some of the principal results of [49] needed in this paper. Lemma 2.2.
Let z ∈ C \{ } and suppose { u ( z, k ) } k ∈ Z , { v ( z, k ) } k ∈ Z ∈ s( Z ) . Then the followingitems ( i ) – ( iii ) are equivalent: ( i ) ( U u ( z, · ))( k ) = zu ( z, k ) , ( W u ( z, · ))( k ) = zv ( z, k ) , k ∈ Z . (2.11)( ii ) ( W u ( z, · ))( k ) = zv ( z, k ) , ( V v ( z, · ))( k ) = u ( z, k ) , k ∈ Z . (2.12)( iii ) (cid:18) u ( z, k ) v ( z, k ) (cid:19) = T ( z, k ) (cid:18) u ( z, k − v ( z, k − (cid:19) , k ∈ Z , (2.13) where the transfer matrices T ( z, k ) , z ∈ C \{ } , k ∈ Z , are given by T ( z, k ) = ρ k α k z /z α k ! , k odd, ρ k α k α k ! , k even. (2.14) Here U , V , and W are understood in the sense of difference expressions on s( Z ) rather than differenceoperators on ℓ ( Z ) . If one sets α k = e it , t ∈ [0 , π ), for some reference point k ∈ Z , then the CMV operator (denotedin this case by U ( t ) k ) splits into a direct sum of two half-lattice operators U ( t ) − ,k − and U ( t )+ ,k actingon ℓ (( −∞ , k − ∩ Z ) and on ℓ ([ k , ∞ ) ∩ Z ), respectively. Explicitly, one obtains U ( t ) k = U ( t ) − ,k − ⊕ U ( t )+ ,k on ℓ (( −∞ , k − ∩ Z ) ⊕ ℓ ([ k , ∞ ) ∩ Z )if α k = e it , t ∈ [0 , π ) . (2.15)(Strictly, speaking, setting α k = e it , t ∈ [0 , π ), for some reference point k ∈ Z contradicts ourbasic Hypothesis 2.1. However, as long as the exception to Hypothesis 2.1 refers to only one site, INIMAL RANK DECOUPLING OF CMV OPERATORS 5 we will safely ignore this inconsistency in favor of the notational simplicity it provides by avoidingthe introduction of a properly modified hypothesis on { α k } k ∈ Z .) Similarly, one obtains V ( t ) k , W ( t ) k , V ( t ) ± ,k , and W ( t ) ± ,k , so that V ( t ) k = V ( t ) − ,k − ⊕ V ( t )+ ,k , W ( t ) k = W ( t ) − ,k − ⊕ W ( t )+ ,k ,U ( t ) k = V ( t ) k W ( t ) k , U ( t ) ± ,k = V ( t ) ± ,k W ( t ) ± ,k . (2.16)For simplicity we will abbreviate U ± ,k = U ( t =0) ± ,k = V ( t =0) ± ,k W ( t =0) ± ,k = V ± ,k W ± ,k . (2.17)It is instructive to introduce one more sequence of Verblunsky coefficients β = { β k } k ∈ Z ∈ s( Z ) by β k = e − it α k , k ∈ Z , (2.18)so that the sequence { ρ k } k ∈ Z is unchanged and α k = e it corresponds to β k = 1. Then the CMVoperators U β and U ± ,k ; β associated with β are unitarily equivalent to the corresponding CMVoperators U α and U ( t ) ± ,k ; α associated with α . Indeed, one verifies that (cid:18) e − it/ e it/ (cid:19) (cid:18) − α k ρ k ρ k α k (cid:19) (cid:18) e − it/ e it/ (cid:19) = (cid:18) − β k ρ k ρ k β k (cid:19) , k ∈ Z , (2.19)and hence, setting A to be the following diagonal unitary operator on ℓ ( Z ), A = e − it/ δ odd + e it/ δ even , (2.20)one obtains for the full-lattice and the direct sum of half-lattice CMV operators, AU α A ∗ = [ AV α A ][ A ∗ W α A ∗ ] = V β W β = U β , (2.21) AU ( t ) k ; α A ∗ = (cid:2) AV ( t ) k ; α A (cid:3)(cid:2) A ∗ W ( t ) k ; α A ∗ (cid:3) = V k ; β W k ; β = U k ; β . (2.22)We refer to [50, Sect. 3] for additional results on CMV operators with Verblunsky coefficients relatedvia (2.18).Now we turn to our principal result of this section. Theorem 2.3.
Fix t , t ∈ [0 , π ) , k ∈ Z , z ∈ C \ ∂ D , and let U ( t ,t ) k denote the following unitaryoperator on ℓ ( Z ) , U ( t ,t ) k = U ( t ) − ,k − ⊕ U ( t )+ ,k . (2.23) Then U − U ( t ,t ) k and ( U − zI ) − − (cid:0) U ( t ,t ) k − zI (cid:1) − are of rank one if and only if the relation t = 2 arg (cid:2) i ( α k e − it / − e it / ) (cid:3) holds. Otherwise, these differences are of rank two. In particular, U − U ( t ) k and ( U − zI ) − − (cid:0) U ( t ) k − zI (cid:1) − are of rank two for any t ∈ [0 , π ) .Proof. Similar to (2.23) we introduce unitary operators V ( t ,t ) k and W ( t ,t ) k by V ( t ,t ) k = V ( t ) − ,k − ⊕ V ( t )+ ,k and W ( t ,t ) k = W ( t ) − ,k − ⊕ W ( t )+ ,k on ℓ ( Z ) . (2.24)Then U ( t ,t ) k = V ( t ,t ) k W ( t ,t ) k by (2.16), and hence, it follows from (2.5) that U − U ( t ,t ) k = ( V (cid:0) W − W ( t ,t ) k (cid:1) , k odd, (cid:0) V − V ( t ,t ) k (cid:1) W, k even. (2.25) S. CLARK, F. GESZTESY, AND M. ZINCHENKO
For k odd, D = W − W ( t ,t ) k is block-diagonal with all its 2 × (cid:18) D k − ,k − D k − ,k D k ,k − D k ,k (cid:19) = (cid:18) − α k ρ ρ k α k (cid:19) − (cid:18) − e it e − it (cid:19) . (2.26)Thus, the difference U − U ( t ,t ) k in (2.25) is always of rank one or two and it is precisely of rank oneif and only if the 2 × (cid:18) D k − ,k − D k − ,k D k ,k − D k ,k (cid:19) = e it α k + e − it α k − e i ( t − t ) −
1= ( α k − e − it ) (cid:18) e it + e − it α k − e it α k − e − it (cid:19) = ( α k − e − it ) e it − iα k e − it / − e it / iα k e − it / − e it / ! , (2.27)which holds if and only if t = 2 arg (cid:2) i ( α k e − it / − e it / ) (cid:3) . The case of even k follows similarly.Finally, the statement for the resolvents follows from the result for U − U ( t ,t ) k and the followingidentity,( U − zI ) − − ( U ( t ,t ) k − zI ) − = − ( U − zI ) − h U − U ( t ,t ) k i ( U ( t ,t ) k − zI ) − . (2.28) (cid:3) Next we present formulas that link various spectral theoretic objects associated with half-latticeCMV operators U ( t ) ± ,k for different values of t ∈ [0 , π ). We start with an analog of Lemma 2.2 fordifference expressions U ( t ) ± ,k , V ( t ) ± ,k , and W ( t ) ± ,k . In the special case t = 0 it is proven in [49, Lem.2.3] and the general case below follows immediately from the special case and the observation ofunitary equivalence in (2.22). Lemma 2.4.
Fix t ∈ [0 , π ) , k ∈ Z , z ∈ C \{ } , and let (cid:8) ˆ p ( t )+ ( z, k, k ) (cid:9) k ≥ k , (cid:8) ˆ r ( t )+ ( z, k, k ) (cid:9) k ≥ k ∈ s([ k , ∞ ) ∩ Z ) . Then the following items ( i ) – ( iii ) are equivalent: ( i ) (cid:0) U ( t )+ ,k ˆ p ( t )+ ( z, · , k ) (cid:1) ( k ) = z ˆ p ( t )+ ( z, k, k ) , (cid:0) W ( t )+ ,k ˆ p ( t )+ ( z, · , k ) (cid:1) ( k ) = z ˆ r ( t )+ ( z, k, k ) , k ≥ k . (2.29)( ii ) (cid:0) W ( t )+ ,k ˆ p ( t )+ ( z, · , k ) (cid:1) ( k ) = z ˆ r ( t )+ ( z, k, k ) , (cid:0) V ( t )+ ,k ˆ r ( t )+ ( z, · , k ) (cid:1) ( k ) = ˆ p ( t )+ ( z, k, k ) , k ≥ k . (2.30)( iii ) (cid:18) ˆ p ( t )+ ( z, k, k )ˆ r ( t )+ ( z, k, k ) (cid:19) = T ( z, k ) (cid:18) ˆ p ( t )+ ( z, k − , k )ˆ r ( t )+ ( z, k − , k ) (cid:19) , k > k , ˆ p ( t )+ ( z, k , k ) = ( ze it ˆ r ( t )+ ( z, k , k ) , k odd ,e − it ˆ r ( t )+ ( z, k , k ) , k even . (2.31) INIMAL RANK DECOUPLING OF CMV OPERATORS 7
Similarly, let (cid:8) ˆ p ( t ) − ( z, k, k ) (cid:9) k ≤ k , (cid:8) ˆ r ( t ) − ( z, k, k ) (cid:9) k ≤ k ∈ s(( −∞ , k ] ∩ Z ) . Then the following items ( iv ) – ( vi ) are equivalent: ( iv ) (cid:0) U ( t ) − ,k ˆ p ( t ) − ( z, · , k ) (cid:1) ( k ) = z ˆ p ( t ) − ( z, k, k ) , (cid:0) W ( t ) − ,k ˆ p ( t ) − ( z, · , k ) (cid:1) ( k ) = z ˆ r ( t ) − ( z, k, k ) , k ≤ k . (2.32)( v ) (cid:0) W ( t ) − ,k ˆ p ( t ) − ( z, · , k ) (cid:1) ( k ) = z ˆ r ( t ) − ( z, k, k ) , (cid:0) V ( t ) − ,k ˆ r ( t ) − ( z, · , k ) (cid:1) ( k ) = ˆ p ( t ) − ( z, k, k ) , k ≤ k . (2.33)( vi ) (cid:18) ˆ p ( t ) − ( z, k − , k )ˆ r ( t ) − ( z, k − , k ) (cid:19) = T ( z, k ) − (cid:18) ˆ p ( t ) − ( z, k, k )ˆ r ( t ) − ( z, k, k ) (cid:19) , k ≤ k , ˆ p ( t ) − ( z, k , k ) = ( − e it ˆ r ( t ) − ( z, k , k ) , k odd, − ze − it ˆ r ( t ) − ( z, k , k ) , k even. (2.34)In the following, we denote by (cid:16) p ( t ) ± ( z,k,k ) r ( t ) ± ( z,k,k ) (cid:17) k ∈ Z and (cid:16) q ( t ) ± ( z,k,k ) s ( t ) ± ( z,k,k ) (cid:17) k ∈ Z , z ∈ C \{ } , four linearlyindependent solutions of (2.13) with the initial conditions: (cid:18) p ( t )+ ( z, k , k ) r ( t )+ ( z, k , k ) (cid:19) = ((cid:0) zγγ (cid:1) , k odd, (cid:0) γγ (cid:1) , k even, (cid:18) q ( t )+ ( z, k , k ) s ( t )+ ( z, k , k ) (cid:19) = ((cid:0) zγ − γ (cid:1) , k odd, (cid:0) − γγ (cid:1) , k even. (2.35) (cid:18) p ( t ) − ( z, k , k ) r ( t ) − ( z, k , k ) (cid:19) = ((cid:0) γ − γ (cid:1) , k odd, (cid:0) − zγγ (cid:1) , k even, (cid:18) q ( t ) − ( z, k , k ) s ( t ) − ( z, k , k ) (cid:19) = ((cid:0) γγ (cid:1) , k odd, (cid:0) zγγ (cid:1) , k even, (2.36)where γ = e − it/ . Then it follows that p ( t ) ± ( z, k, k ), q ( t ) ± ( z, k, k ), r ( t ) ± ( z, k, k ), and s ( t ) ± ( z, k, k ), k, k ∈ Z , are Laurent polynomials in z , that is, finite linear combinations of terms z k , k ∈ Z , withcomplex-valued coefficients.Since all of the above sequences satisfy the same recursion relation (2.13) which can have at mosttwo linearly independent solutions, these sequences satisfy various identities. Some of them we statein the following lemma. S. CLARK, F. GESZTESY, AND M. ZINCHENKO
Lemma 2.5.
Let t , t ∈ [0 , π ) and γ j = e − it j / , j = 1 , . Then (cid:18) q ( t ) ± ( z, · , k ) s ( t ) ± ( z, · , k ) (cid:19) = Re( γ γ ) (cid:18) q ( t ) ± ( z, · , k ) s ( t ) ± ( z, · , k ) (cid:19) + i Im( γ γ ) (cid:18) p ( t ) ± ( z, · , k ) r ( t ) ± ( z, · , k ) (cid:19) , (2.37) (cid:18) p ( t ) ± ( z, · , k ) r ( t ) ± ( z, · , k ) (cid:19) = i Im( γ γ ) (cid:18) q ( t ) ± ( z, · , k ) s ( t ) ± ( z, · , k ) (cid:19) + Re( γ γ ) (cid:18) p ( t ) ± ( z, · , k ) r ( t ) ± ( z, · , k ) (cid:19) , (2.38) (cid:18) q ( t ) − ( z, · , k ) s ( t ) − ( z, · , k ) (cid:19) = γ γ − γ γ z z k (mod 2) (cid:18) q ( t )+ ( z, · , k ) s ( t )+ ( z, · , k ) (cid:19) + γ γ + γ γ z z k (mod 2) (cid:18) p ( t )+ ( z, · , k ) r ( t )+ ( z, · , k ) (cid:19) , (2.39) (cid:18) p ( t ) − ( z, · , k ) r ( t ) − ( z, · , k ) (cid:19) = γ γ + γ γ z z k (mod 2) (cid:18) q ( t )+ ( z, · , k ) s ( t )+ ( z, · , k ) (cid:19) + γ γ − γ γ z z k (mod 2) (cid:18) p ( t )+ ( z, · , k ) r ( t )+ ( z, · , k ) (cid:19) , (2.40) (cid:18) q ( t ) − ( z, · , k − s ( t ) − ( z, · , k − (cid:19) = i Im( γ γ + α k γ γ ) ρ k (cid:18) q ( t )+ ( z, · , k ) s ( t )+ ( z, · , k ) (cid:19) + Re( γ γ + α k γ γ ) ρ k (cid:18) p ( t )+ ( z, · , k ) r ( t )+ ( z, · , k ) (cid:19) , (2.41) (cid:18) p ( t ) − ( z, · , k − r ( t ) − ( z, · , k − (cid:19) = Re( γ γ − α k γ γ ) ρ k (cid:18) q ( t )+ ( z, · , k ) s ( t )+ ( z, · , k ) (cid:19) + i Im( γ γ − α k γ γ ) ρ k (cid:18) p ( t )+ ( z, · , k ) r ( t )+ ( z, · , k ) (cid:19) . (2.42) In particular, whenever t = 2 arg (cid:2) i ( α k e − it / − e it / ) (cid:3) identity (2.42) simplifies to (cid:18) p ( t ) − ( z, · , k − r ( t ) − ( z, · , k − (cid:19) = i (cid:12)(cid:12) e it − α k (cid:12)(cid:12) ρ k (cid:18) p ( t )+ ( z, · , k ) r ( t )+ ( z, · , k ) (cid:19) . (2.43) Proof.
Since both sides of (2.37)–(2.42) satisfy the same recursion relation (2.13) it suffices to checkthese equalities only at one point, say at point k = k . Substituting (2.35) and (2.36) into (2.37)–(2.40) one verifies the first four identities. Using (2.35) and (2.36) once again and applying transfermatrix T ( z, k ) to the left hand-sides of (2.41)–(2.43) one verifies the last three identities. (cid:3) Next, following [49] we introduce half-lattice Weyl–Titchmarsh m -functions associated with theCMV operators U ( t ) ± ,k by m ( t ) ± ( z, k ) = ± ( δ k , ( U ( t ) ± ,k + zI )( U ( t ) ± ,k − zI ) − δ k ) ℓ ( Z ∩ [ k , ±∞ )) = ± I ∂ D dµ ( t ) ± ( ζ, k ) ζ + zζ − z , z ∈ C \ ∂ D , (2.44)where dµ ( t ) ± ( ζ, k ) = d ( δ k , E U ( t ) ± ,k ( ζ ) δ k ) ℓ ( Z ∩ [ k , ±∞ )) , ζ ∈ ∂ D , (2.45)and dE U ( t ) ± ,k ( · ) denote the operator-valued spectral measures of the operators U ( t ) ± ,k , U ( t ) ± ,k = I ∂ D dE U ( t ) ± ,k ( ζ ) ζ. (2.46) INIMAL RANK DECOUPLING OF CMV OPERATORS 9
Then following the steps of [49, Cor. 2.14] one verifies that (cid:18) q ( t ) ± ( z, · , k ) s ( t ) ± ( z, · , k ) (cid:19) + m ( t ) ± ( z, k ) (cid:18) p ( t ) ± ( z, · , k ) r ( t ) ± ( z, · , k ) (cid:19) ∈ ℓ ([ k , ±∞ ) ∩ Z ) , z ∈ C \ ( ∂ D ∪ { } ) . (2.47)The special case t = 0 of the next result is proven in [49, Cor. 2.16 and Thm. 2.18]. The generalcase of t ∈ [0 , π ) stated below follows along the same lines and hence we omit the details for brevity. Theorem 2.6.
Let t ∈ [0 , π ) and k ∈ Z . Then there exist unique Caratheodory ( resp. anti-Caratheodory ) functions M ( t ) ± ( · , k ) such that (cid:18) u ( t ) ± ( z, · , k ) v ( t ) ± ( z, · , k ) (cid:19) = (cid:18) q ( t )+ ( z, · , k ) s ( t )+ ( z, · , k ) (cid:19) + M ( t ) ± ( z, k ) (cid:18) p ( t )+ ( z, · , k ) r ( t )+ ( z, · , k ) (cid:19) ∈ ℓ ([ k , ±∞ ) ∩ Z ) ,z ∈ C \ ( ∂ D ∪ { } ) . (2.48) In addition, sequence (cid:16) u ( t ) ± ( z,k,k ) v ( t ) ± ( z,k,k ) (cid:17) k ∈ Z satisfies (2.13) and is unique ( up to constant scalar multiples ) among all sequence that satisfy (2.13) and are square summable near ±∞ . We will call u ( t ) ± ( z, · , k ) and v ( t ) ± ( z, · , k ) Weyl–Titchmarsh solutions of U . Similarly, we willcall m ( t ) ± ( z, k ) as well as M ( t ) ± ( z, k ) the half-lattice Weyl–Titchmarsh m -functions associated with U ( t ) ± ,k . (See also [69] for a comparison of various alternative notions of Weyl–Titchmarsh m -functionsfor U + ,k .)Using (2.39)–(2.42), (2.47), and Theorem 2.6 one also verifies that M ( t )+ ( z, k ) = m ( t )+ ( z, k ) , z ∈ C \ ∂ D , (2.49) M ( t )+ (0 , k ) = 1 , (2.50) M ( t ) − ( z, k ) = Re( a k ) + i Im( b k ) m ( t ) − ( z, k − i Im( a k ) + Re( b k ) m ( t ) − ( z, k −
1) = (1 − z ) m ( t ) − ( z, k ) + (1 + z )(1 + z ) m ( t ) − ( z, k ) + (1 − z )= (cid:0) m ( t ) − ( z, k ) + 1 (cid:1) − z (cid:0) m ( t ) − ( z, k ) − (cid:1)(cid:0) m ( t ) − ( z, k ) + 1 (cid:1) + z (cid:0) m ( t ) − ( z, k ) − (cid:1) , z ∈ C \ ∂ D , (2.51) M ( t ) − (0 , k ) = α k + e it α k − e it , (2.52) m ( t ) − ( z, k ) = Re( a k +1 ) − i Im( a k +1 ) M ( t ) − ( z, k + 1)Re( b k +1 ) M ( t ) − ( z, k + 1) − i Im( b k +1 )= z (cid:0) M ( t ) − ( z, k ) + 1 (cid:1) − (cid:0) M ( t ) − ( z, k ) − (cid:1) z (cid:0) M ( t ) − ( z, k ) + 1 (cid:1) + (cid:0) M ( t ) − ( z, k ) − (cid:1) , z ∈ C \ ∂ D , (2.53)where a k = 1 + e − it α k and b k = 1 − e − it α k , k ∈ Z . In particular, one infers that M ( t ) ± are analyticat z = 0.Next, we introduce the Schur (resp. anti-Schur) functions Φ ( t ) ± ( · , k ), k ∈ Z , byΦ ( t ) ± ( z, k ) = M ( t ) ± ( z, k ) − M ( t ) ± ( z, k ) + 1 , z ∈ C \ ∂ D . (2.54) Then by (2.53) and (2.54), M ( t ) ± ( z, k ) = 1 + Φ ( t ) ± ( z, k )1 − Φ ( t ) ± ( z, k ) , m ( t ) − ( z, k ) = z − Φ ( t ) − ( z, k ) z + Φ ( t ) − ( z, k ) , z ∈ C \ ∂ D . (2.55)Moreover, it follows from (2.35), (2.54), and Theorem 2.6 thatΦ ( t ) ± ( z, k ) = ze it v ( t ) ± ( z,k,k ) u ( t ) ± ( z,k,k ) , k odd, e it u ( t ) ± ( z,k,k ) v ( t ) ± ( z,k,k ) , k even, k ∈ Z , z ∈ C \ ∂ D , (2.56)where u ( t ) ± ( · , k, k ) and v ( t ) ± ( · , k, k ) are the sequences defined in (2.48). Since the Weyl–Titchmarshsolution (cid:16) u ( t ) ± ( z,k,k ) v ( t ) ± ( z,k,k ) (cid:17) k ∈ Z is unique up to a multiplicative constant, we conclude from (2.56) that e − it Φ ( t ) ± ( · , k ) is actually t -independent. Thus, fixing t , t ∈ [0 , π ), one computesΦ ( t ) ± ( · , k ) = e i ( t − t ) Φ ( t ) ± ( · , k ) , (2.57) M ( t ) ± ( · , k ) = i Im( e i ( t − t ) / ) + Re( e i ( t − t ) / ) M ( t ) ± ( · , k )Re( e i ( t − t ) / ) + i Im( e i ( t − t ) / ) M ( t ) ± ( · , k ) , k ∈ Z . (2.58)Finally, following [49], we obtain the following identities, r ( t )+ ( z, k, k ) = z k (mod 2) p ( t )+ (1 /z, k, k ) , (2.59) s ( t )+ ( z, k, k ) = − z k (mod 2) q ( t )+ (1 /z, k, k ) , (2.60) r ( t ) − ( z, k, k ) = − z ( k +1) (mod 2) p ( t ) − (1 /z, k, k ) , (2.61) s ( t ) − ( z, k, k ) = z ( k +1) (mod 2) q ( t ) − (1 /z, k, k ) , (2.62)and provide formulas for the resolvents of the half-lattice CMV operators U ( t ) ± ,k and the full-latticeCMV operator U . In the special case t = 0 these formulas were obtained in (2.63)–(2.66), (2.171),(2.172), and (3.7) of [49]. INIMAL RANK DECOUPLING OF CMV OPERATORS 11
Lemma 2.7.
Let t ∈ [0 , π ) , k ∈ Z , and z ∈ C \ ( ∂ D ∪ { } ) . Then the resolvent (cid:0) U ( t ) ± ,k − zI (cid:1) − isgiven in terms of its matrix representation in the standard basis of ℓ ([ k , ±∞ ) ∩ Z ) by (cid:0) U ( t )+ ,k − zI (cid:1) − ( k, k ′ ) = z − k (mod 2) z × ( p ( t )+ ( z, k, k ) b v ( t )+ ( z, k ′ , k ) , k < k ′ and k = k ′ odd, b u ( t )+ ( z, k, k ) r ( t )+ ( z, k ′ , k ) , k > k ′ and k = k ′ even (2.63)= 12 z − p ( t )+ ( z, k, k ) b u ( t )+ (1 /z, k ′ , k ) , k < k ′ and k = k ′ odd, b u ( t )+ ( z, k, k ) p ( t )+ (1 /z, k ′ , k ) , k > k ′ and k = k ′ even, (2.64) k, k ′ ∈ [ k , ∞ ) ∩ Z , (cid:0) U ( t ) − ,k − zI (cid:1) − ( k, k ′ ) = z − ( k +1) (mod 2) z × (b u ( t ) − ( z, k, k ) r ( t ) − ( z, k ′ , k ) , k < k ′ and k = k ′ odd, p ( t ) − ( z, k, k ) b v ( t ) − ( z, k ′ , k ) , k > k ′ and k = k ′ even (2.65)= 12 z − b u ( t ) − ( z, k, k ) p ( t ) − (1 /z, k ′ , k ) , k < k ′ and k = k ′ odd, p ( t ) − ( z, k, k ) b u ( t ) − (1 /z, k ′ , k ) , k > k ′ and k = k ′ even, (2.66) k, k ′ ∈ ( −∞ , k ] ∩ Z , where the sequences b u ( t ) ± and b v ( t ) ± are defined by (cid:18)b u ( t ) ± ( z, · , k ) b v ( t ) ± ( z, · , k ) (cid:19) = (cid:18) q ( t ) − ( z, · , k ) s ( t ) − ( z, · , k ) (cid:19) + m ( t ) ± ( z, k ) (cid:18) p ( t ) − ( z, · , k ) r ( t ) − ( z, · , k ) (cid:19) ∈ ℓ ([ k , ±∞ ) ∩ Z ) ,z ∈ C \ ( ∂ D ∪ { } ) . (2.67) The corresponding result for the full-lattice resolvent of U then reads ( U − zI ) − ( k, k ′ ) = − z − k (mod 2) z [ M ( t )+ ( z, k ) − M ( t ) − ( z, k )] × ( u ( t ) − ( z, k, k ) v ( t )+ ( z, k ′ , k ) , k < k ′ and k = k ′ odd, u ( t )+ ( z, k, k ) v ( t ) − ( z, k ′ , k ) , k > k ′ and k = k ′ even (2.68)= u ( t ) − ( z, k, k ) u ( t )+ (1 /z, k ′ , k ) , k < k ′ and k = k ′ odd, u ( t )+ ( z, k, k ) u ( t ) − (1 /z, k ′ , k ) , k > k ′ and k = k ′ even, z [ M ( t )+ ( z, k ) − M ( t ) − ( z, k )] k, k ′ ∈ Z . (2.69) Moreover, since U ( t ) ± ,k and U are unitary and hence zero is in the resolvent set, (2.63) – (2.69) ana-lytically extend to z = 0 . CMV operators with matrix-valued coefficients
In the remainder of this paper, C m × m denotes the space of m × m matrices with complex-valuedentries endowed with the operator norm k·k C m × m (we use the standard Euclidean norm in C m ). Theadjoint of an element γ ∈ C m × m is denoted by γ ∗ , I m denotes the identity matrix in C m , and thereal and imaginary parts of γ are defined as usual by Re( γ ) = ( γ + γ ∗ ) / γ ) = ( γ − γ ∗ ) / (2 i ). Remark . For simplicity of exposition, we find it convenient to use the following conventions: Wedenote by s( Z ) the vector space of all C -valued sequences, and by s( Z ) m = s( Z ) ⊗ C m the vectorspace of all C m -valued sequences; that is, φ = { φ ( k ) } k ∈ Z = ... φ ( − φ (0) φ (1)... ∈ s( Z ) m , φ ( k ) = ( φ ( k )) ( φ ( k )) ...( φ ( k )) m ∈ C m , k ∈ Z . (3.1)Moreover, we introduce s( Z ) m × n = s( Z ) m ⊗ C n , m, n ∈ N , that is, Φ = ( φ , . . . , φ n ) ∈ s( Z ) m × n ,where φ j ∈ s( Z ) m for all j = 1 , . . . , n .We also note that s( Z ) m × n = s( Z ) ⊗ C m × n , m, n ∈ N ; which is to say that the elements of s( Z ) m × n can be identified with the C m × n -valued sequences,Φ = { Φ( k ) } k ∈ Z = ...Φ( − , Φ( k ) = (Φ( k )) , . . . (Φ( k )) ,n ... ...(Φ( k )) m, . . . (Φ( k )) m,n ∈ C m × n , k ∈ Z , (3.2)by setting Φ = ( φ , . . . , φ n ), where φ j = ... φ j ( − φ j (0) φ j (1)... ∈ s( Z ) m , φ j ( k ) = (Φ( k )) ,j ...(Φ( k )) m,j ∈ C m , j = 1 , . . . , n, k ∈ Z . (3.3)For the elements of s( Z ) m × n we define the right-multiplication by n × n matrices with complex-valued entries byΦ C = ( φ , . . . , φ n ) c , . . . c ,n ... ... c n, . . . c n,n = n X j =1 φ j c j, , . . . , n X j =1 φ j c j,n ∈ s( Z ) m × n (3.4)for all Φ ∈ s( Z ) m × n and C ∈ C n × n . In addition, for any linear transformation A : s( Z ) m → s( Z ) m ,we define A Φ for all Φ = ( φ , . . . , φ n ) ∈ s( Z ) m × n by A Φ = ( A φ , . . . , A φ n ) ∈ s( Z ) m × n . (3.5)Given the above conventions, we note the subspace containment: ℓ ( Z ) m = ℓ ( Z ) ⊗ C m ⊂ s( Z ) m and ℓ ( Z ) m × n = ℓ ( Z ) ⊗ C m × n ⊂ s( Z ) m × n . We also note that ℓ ( Z ) m represents a Hilbert spacewith scalar product given by( φ, ψ ) ℓ ( Z ) m = ∞ X k = −∞ m X j =1 ( φ ( k )) j ( ψ ( k )) j , φ, ψ ∈ ℓ ( Z ) m . (3.6)Finally, we note that a straightforward modification of the above definitions also yields the Hilbertspace ℓ ( J ) m as well as the sets ℓ ( J ) m × n , s( J ) m , and s( J ) m × n for any J ⊂ Z .We start by introducing our basic assumption: INIMAL RANK DECOUPLING OF CMV OPERATORS 13
Hypothesis 3.2.
Let m ∈ N and assume α = { α k } k ∈ Z is a sequence of m × m matrices with complexentries and such that k α k k C m × m < , k ∈ Z . (3.7)Given a sequence α satisfying (3.7), we define two sequences of positive self-adjoint m × m matrices { ρ k } k ∈ Z and { e ρ k } k ∈ Z by ρ k = [ I m − α ∗ k α k ] / , k ∈ Z , (3.8) e ρ k = [ I m − α k α ∗ k ] / , k ∈ Z , (3.9)and two sequences of m × m matrices with positive real parts, { a k } k ∈ Z ⊂ C m × m and { b k } k ∈ Z ⊂ C m × m by a k = I m + α k , k ∈ Z , (3.10) b k = I m − α k , k ∈ Z . (3.11)Then (3.7) implies that ρ k and e ρ k are invertible matrices for all k ∈ Z , and using elementary powerseries expansions one verifies the following identities for all k ∈ Z , e ρ ± k α k = α k ρ ± k , α ∗ k e ρ ± k = ρ ± k α ∗ k , a ∗ k e ρ − k a k = a k ρ − k a ∗ k , (3.12) b ∗ k e ρ − k b k = b k ρ − k b ∗ k , a ∗ k e ρ − k b k + a k ρ − k b ∗ k = b ∗ k e ρ − k a k + b k ρ − k a ∗ k = 2 I m . (3.13)According to Simon [71], we call α k the Verblunsky coefficients in honor of Verblunsky’s pioneeringwork in the theory of orthogonal polynomials on the unit circle [80], [81].Next, we introduce a sequence of 2 × k with m × m matrix coefficientsby Θ k = (cid:18) − α k e ρ k ρ k α ∗ k (cid:19) , k ∈ Z , (3.14)and two unitary operators V and W on ℓ ( Z ) m by their matrix representations in the standard basisof ℓ ( Z ) m by V = . . . Θ k − Θ k . . . , W = . . . Θ k − Θ k +1 . . . , (3.15)where (cid:18) V k − , k − V k − , k V k, k − V k, k (cid:19) = Θ k , (cid:18) W k, k W k, k +1 W k +1 , k W k +1 , k +1 (cid:19) = Θ k +1 , k ∈ Z . (3.16)Moreover, we introduce the unitary operator U on ℓ ( Z ) m as the product of the unitary operators V and W by U = VW , (3.17) We emphasize that α k ∈ C m × m , k ∈ Z , are general (not necessarily normal) matrices. or in matrix form in the standard basis of ℓ ( Z ) m , by U = . . . . . . . . . . . . . . . − α ρ − − α α ∗− − e ρ α e ρ e ρ ρ ρ − ρ α ∗− − α ∗ α α ∗ e ρ − α ρ − α α ∗ − e ρ α e ρ e ρ ρ ρ ρ α ∗ − α ∗ α α ∗ e ρ
0. . . . . . . . . . . . . . . . (3.18)Here terms of the form − α k α ∗ k − and − α ∗ k α k +1 , k ∈ Z , represent the diagonal entries U k − , k − and U k, k of the infinite matrix U in (3.18), respectively. Then, with δ even and δ odd defined in (2.10),and by analogy with (2.9), we see that as an operator acting upon ℓ ( Z ) m , U can be represented by U = ρρ − δ even S −− + [ ρ ( α − ) ∗ δ even − α ∗ ρ δ odd ] S − − α ∗ α + δ even − α + α ∗ δ odd + [ α ∗ e ρ + δ even − e ρ + α ++ δ odd ] S + + e ρ + e ρ ++ δ odd S ++ . (3.19)We continue to call the operator U on ℓ ( Z ) m the CMV operator since (3.14)–(3.18) in the context ofthe scalar-valued semi-infinite (i.e., half-lattice) case were obtained by Cantero, Moral, and Vel´azquezin [17] in 2003. Then, in analogy with Lemma 2.2, the following result is proven in [19]: Lemma 3.3.
Let z ∈ C \{ } and { U ( z, k ) } k ∈ Z , { V ( z, k ) } k ∈ Z be two C m × m -valued sequences. Thenthe following items ( i ) – ( iii ) are equivalent: ( i ) ( U U ( z, · ))( k ) = zU ( z, k ) , ( W U ( z, · ))( k ) = zV ( z, k ) , k ∈ Z . (3.20)( ii ) ( W U ( z, · ))( k ) = zV ( z, k ) , ( V V ( z, · ))( k ) = U ( z, k ) , k ∈ Z . (3.21)( iii ) (cid:18) U ( z, k ) V ( z, k ) (cid:19) = T ( z, k ) (cid:18) U ( z, k − V ( z, k − (cid:19) , k ∈ Z . (3.22) Here U , V , and W are understood in the sense of difference expressions on s( Z ) m × m rather thandifference operators on ℓ ( Z ) m ( cf. Remark and the transfer matrices T ( z, k ) , z ∈ C \{ } , k ∈ Z ,are defined by T ( z, k ) = e ρ − k α k z e ρ − k z − ρ − k ρ − k α ∗ k ! , k odd, ρ − k α ∗ k ρ − k e ρ − k e ρ − k α k ! , k even. (3.23) Definition 3.4.
If for some reference point, k ∈ Z , we modify Hypothesis 3.2 by allowing α k = γ ,where γ ∈ C m × m is unitary, then the resulting operators defined in (3.15)–(3.17) will be denoted by V ( γ ) k , W ( γ ) k and U ( γ ) k . Remark . Strictly, speaking, allowing α k to be unitary for some reference point k ∈ Z contradictsour basic Hypothesis 3.2. However, as long as the exception to Hypothesis 3.2 refers to only one site,we will safely ignore this inconsistency in favor of the notational simplicity it provides by avoidingthe introduction of a properly modified hypothesis on α = { α k } k ∈ Z and will refer to U ( γ ) k as a CMVoperator.The operator U ( γ ) k splits into a direct sum of two half-lattice operators U ( γ ) − ,k − and U ( γ ) − ,k actingon ℓ (( −∞ , k − ∩ Z ) m and on ℓ ([ k , ∞ ) ∩ Z ) m , respectively. Explicitly, one obtains U ( γ ) k = U ( γ ) − ,k − ⊕ U ( γ )+ ,k in ℓ (( −∞ , k − ∩ Z ) m ⊕ ℓ ([ k , ∞ ) ∩ Z ) m . (3.24) INIMAL RANK DECOUPLING OF CMV OPERATORS 15
Similarly, one obtains W ( γ ) − ,k − , V ( γ ) − ,k − and W ( γ )+ ,k , V ( γ )+ ,k such that V ( γ ) k = V ( γ ) − ,k − ⊕ V ( γ )+ ,k , W ( γ ) k = W ( γ ) − ,k − ⊕ W ( γ )+ ,k , (3.25)and hence U ( γ ) ± ,k = V ( γ ) ± ,k W ( γ ) ± ,k , U ( γ ) k = V ( γ ) k W ( γ ) k . (3.26)For the special case when γ = I m , we simplify our notation by writing U k = U ( γ = I m ) k = V ( γ = I m ) k W ( γ = I m ) k = V k W k (3.27) U ± ,k = U ( γ = I m ) ± ,k = V ( γ = I m ) ± ,k W ( γ = I m ) ± ,k = V ± ,k W ± ,k (3.28)In analogy with the scalar case, when emphasizing dependence of these operators on the sequence α = { α k } k ∈ Z , we write, for example, U α and U ( γ ) α ; k . Also in analogy to the scalar case, let σ, τ ∈ C m × m be unitary, and let A and e A be the unitary operators defined on ℓ ( Z ) m by A = σδ odd + τ δ even , e A = τ δ odd + σδ even , (3.29)Then, for the full-lattice and direct sum of half-lattice CMV operators, we obtain the followinganalogs of (2.21) and (2.22): AU β A ∗ = (cid:2) AV β e A ∗ (cid:3)(cid:2)e AW β A ∗ (cid:3) = V α W α = U α , (3.30) AU ( γ ) β ; k A ∗ = (cid:2) AV ( γ ) β ; k e A ∗ (cid:3)(cid:2)e AW ( γ ) β ; k A ∗ (cid:3) = V ( σγτ ∗ ) α ; k W ( σγτ ∗ ) α ; k = U ( σγτ ∗ ) α ; k . (3.31)where β = { β k } k ∈ Z , and α k = σβ k τ ∗ . In particular, when σ = γ − / , τ = γ / , we note, by (3.27)and (3.31), that AU ( γ ) β ; k A ∗ = V α ; k W α ; k = U α ; k . (3.32)The unitary transformations cited in (3.29)–(3.31) are relevant to our next result because of thefollowing observation about n × n complex matrices: Let α ∈ C m × m , where we are not assumingthat α is unitary. Then, α has a (not necessarily unique) polar decomposition, α = U | α | , where U, | α | ∈ C m × m , U is unitary, and | α | = ( α ∗ α ) / ≥ | α | can then be diagonalized: | α | = U ∗ DU , where U , D ∈ C m × m , U is unitary, and D is diagonal. Thus, each α ∈ C m × m has a (not necessarily unique) factorizationof the form α = σβτ, (3.33)where, σ, β, τ ∈ C m × m , where σ and τ are unitary, and where β is diagonal.We now present our principal result on rank m perturbations: Theorem 3.6.
Given U α , fix k ∈ Z and let α k = σ k β k τ ∗ k be a factorization for α k , as describedin (3.33) , where σ k , τ k ∈ C m × m are unitary, and where β k ∈ C m × m is the diagonal matrix β k = diag[ β k , , . . . , β k ,m ] . Let γ j = σ k θ j τ ∗ k , j = 1 , , where θ = diag[ e it , . . . , e it m ] , and θ =diag[ e is , . . . , e is m ] . Let U ( γ ,γ ) α ; k denote the unitary operator acting on ℓ ( Z ) m and defined by U ( γ ,γ ) α ; k = U ( γ ) α ; − ,k − ⊕ U ( γ ) α ;+ ,k . (3.34) Then, for an arbitrary unitary matrix γ ∈ C m × m , the difference U α − U ( γ ) α ; k has rank greater than m , while the differences U α − U ( γ ,γ ) α ; k and ( U α − zI ) − − (cid:0) U ( γ ,γ ) α ; k − zI (cid:1) − are of rank m if and onlyif t j = 2 arg[ i ( β k ,j e − is j / − e is j / )] , j = 1 , . . . , m , and otherwise possess rank greater than m . Proof. U ( γ ) α ; k = V ( γ ) α ; k W ( γ ) α ; k by (3.26), and hence, it follows from (3.15) that U α − U ( γ ) α ; k = ( V α (cid:0) W α − W ( γ ) α ; k (cid:1) , k odd, (cid:0) V α − V ( γ ) α ; k (cid:1) W α , k even. (3.35)For k odd, D = W α − W ( γ ) α ; k is block-diagonal with all of its 2 m × m blocks on the diagonal beingzero except for one which has the following form A = (cid:18) D k − ,k − D k − ,k D k ,k − D k ,k (cid:19) = (cid:18) − α k + γ e ρ k ρ k α ∗ k − γ ∗ (cid:19) (3.36)Note that the following subspaces have only trivial intersection with ker( A ) since ρ k and e ρ k areinvertible matrices, S = (cid:26)(cid:18) ξ (cid:19) ∈ C m | ξ ∈ C m (cid:27) , S = (cid:26)(cid:18) η (cid:19) ∈ C m | η ∈ C m (cid:27) . (3.37)As a consequence, rank( A ) > m . Further note, for any c ∈ C , that (cid:18) ξcξ (cid:19) ∈ ker( A ) (3.38)only when ξ = 0 ∈ C m . To see this, assume that (3.38) holds for some c ∈ C and some ξ = 0 ∈ C m .It follows that (cid:18) (cid:19) = (cid:18) cξξ (cid:19) ∗ A (cid:18) ξcξ (cid:19) = (cid:18) − ¯ cξ ∗ ( α k − γ ) ξ + | c | ξ ∗ e ρ k ξξ ∗ ρξ + cξ ∗ ( α k − γ ) ∗ ξ (cid:19) (3.39)and hence by conjugation in the first line of (3.39) that0 = − cξ ∗ ( α k − γ ) ∗ ξ + | c | ξ ∗ e ρ k ξ, (3.40)0 = ξ ∗ ρξ + cξ ∗ ( α k − γ ) ∗ ξ. (3.41)Summing these equations, we see that ξ ∗ ( ρ k + | c | e ρ k ) ξ = 0. However, strict positivity of theself-adjoint matrix ρ k + | c | e ρ k implies that ξ = 0 ∈ C m ; a contradiction.Noting again that rank( A ) > m , assume that rank( A ) = m . Given that C m = S ⊕ S , ker( A ) ∩ S j = { } , j = 1 , , (3.42)then there exists a matrix M ∈ C m × m such thatker( A ) = (cid:26)(cid:18) ξM ξ (cid:19) | ξ ∈ C m (cid:27) . (3.43)However, this implies the existence of some ξ = 0 ∈ C m and some c ∈ C such that (3.38) holds;thus, again a contradiction. Hence, the rank of A , and as a consequence the rank of U α − U ( γ ) α ; k , aregreater than m . The proof when k ∈ Z is even is similar to that just given.Now consider the rank of the difference U α − U ( γ ,γ ) α ; k , first noting by (3.25) and (3.26) that U ( γ ,γ ) α ; k = V ( γ ,γ ) α ; k W ( γ ,γ ) α ; k and hence, by (3.15) that U α − U ( γ ,γ ) α ; k = ( V α (cid:0) W α − W ( γ ,γ ) α ; k (cid:1) , k odd, (cid:0) V α − V ( γ ,γ ) α ; k (cid:1) W α , k even. (3.44)We again proceed by assuming that k ∈ Z is odd, and letting D = W α − W ( γ ,γ ) α ; k be the block-diagonal matrix all of whose 2 m × m blocks on the diagonal are zero except for one which has the INIMAL RANK DECOUPLING OF CMV OPERATORS 17 following form: (cid:18) D k − ,k − D k − ,k D k ,k − D k ,k (cid:19) = (cid:18) − α k e ρ k ρ k α ∗ k (cid:19) − (cid:18) − σ k θ τ ∗ k τ k θ ∗ σ ∗ k (cid:19) (3.45)= (cid:18) σ k τ k (cid:19) (cid:18) − β k + θ κ k κ k β ∗ k − θ ∗ (cid:19) (cid:18) τ ∗ k σ ∗ k (cid:19) , (3.46)where κ k = ( I m − β k β ∗ k ) / = ( I m − β ∗ k β k ) / = diag[ κ k , , . . . , κ k ,m ]. Then, the rank of thedifference D = W α − W ( γ ,γ ) α ; k equals the rank of the matrix B = (cid:18) − β k + θ κ k κ k β ∗ k − θ ∗ (cid:19) . (3.47)Since all m × m matrices on the right-hand side of (3.47) are diagonal, performing an equivalentset of permutations on the rows and on the columns of the matrix B in (3.47) results in a blockdiagonal matrix with 2 × B j = (cid:18) − β k ,j + e it j κ k ,j κ k ,j β k ,j − e − is j (cid:19) , j = 1 , . . . , m. (3.48)It follows that rank( B ) = m precisely when rank( B j ) = 1 for all j = 1 , . . . , m , and otherwise,that rank( B ) > m . We now observe that each B j has the form given in (2.26). Hence, by thederivation following from (2.27) we see that det( B j ) = 0, and hence that rank( B j ) = 1, preciselywhen t j = 2 arg[ i ( β k ,j e − is j / − e is j / )] . As in the scalar case discussed in Theorem 2.3, the statement in this theorem for the resolventsfollows from the result just proven for the difference U α − U ( γ ,γ ) α ; k and from the identity,( U α − zI ) − − (cid:0) U ( γ ,γ ) α ; k − zI (cid:1) − = − ( U α − zI ) − (cid:2) U α − U ( γ ,γ ) α ; k (cid:3)(cid:0) U ( γ ,γ ) α ; k − zI (cid:1) − . (3.49) (cid:3) Remark . In particular, we note that the difference (cid:2) U α − U ( γ ) α ; k (cid:3) has rank greater than m when γ = I m .Next we present formulas for different unitary γ ∈ C m × m , linking various spectral theoreticobjects associated with half-lattice CMV operators U ( γ ) ± ,k . For the special case when γ = I m , theseobjects, and the relationships described below, have proven exceptionally useful (see, e.g., [19], [20],[49], [50], and [87]).We begin with an analog of Lemma 3.3 for difference expressions U ( γ ) ± ,k , V ( γ ) ± ,k , and W ( γ ) ± ,k . Forthe special case γ = I m , the result below is proven in [19, Lemma 2.4]. The general case belowfollows immediately from the special case and the observation of unitary equivalence in (3.32). Lemma 3.8.
Let z ∈ C \{ } , k ∈ Z , and let γ ∈ C m × m be unitary. Let (cid:8) b P ( γ )+ ( z, k, k ) (cid:9) k ≥ k , (cid:8) b R ( γ )+ ( z, k, k ) (cid:9) k ≥ k be two C m × m -valued sequences. Then the following items ( i ) – ( iii ) are equiva-lent: ( i ) (cid:0) U ( γ )+ ,k b P ( γ )+ ( z, · , k ) (cid:1) ( k ) = z b P ( γ )+ ( z, k, k ) , (cid:0) W ( γ )+ ,k b P ( γ )+ ( z, · , k ) (cid:1) ( k ) = z b R ( γ )+ ( z, k, k ) , k ≥ k . (3.50)( ii ) (cid:0) W ( γ )+ ,k b P ( γ )+ ( z, · , k ) (cid:1) ( k ) = z b R ( γ )+ ( z, k, k ) , (cid:0) V ( γ )+ ,k b R ( γ )+ ( z, · , k ) (cid:1) ( k ) = b P ( γ )+ ( z, k, k ) , k ≥ k . (3.51)( iii ) (cid:18) b P ( γ )+ ( z, k, k ) b R ( γ )+ ( z, k, k ) (cid:19) = T ( z, k ) (cid:18) b P ( γ )+ ( z, k − , k ) b R ( γ )+ ( z, k − , k ) (cid:19) , k > k , b P ( γ )+ ( z, k , k ) = ( zγ b R ( γ )+ ( z, k , k ) , k odd ,γ ∗ b R ( γ )+ ( z, k , k ) , k even . (3.52) Similarly, let (cid:8) b P ( γ ) − ( z, k, k ) (cid:9) k ≤ k , (cid:8) b R ( γ ) − ( z, k, k ) (cid:9) k ≤ k be two C m × m -valued sequences. Then thefollowing items ( iv ) – ( vi ) are equivalent: ( iv ) (cid:0) U ( γ ) − ,k b P ( γ ) − ( z, · , k ) (cid:1) ( k ) = z b P ( γ ) − ( z, k, k ) , (cid:0) W ( γ ) − ,k b P ( γ ) − ( z, · , k ) (cid:1) ( k ) = z b R ( γ ) − ( z, k, k ) , k ≤ k . (3.53)( v ) (cid:0) W ( γ ) − ,k b P ( γ ) − ( z, · , k ) (cid:1) ( k ) = z b R ( γ ) − ( z, k, k ) , (cid:0) V ( γ ) − ,k b R ( γ ) − ( z, · , k ) (cid:1) ( k ) = b P ( γ ) − ( z, k, k ) , k ≤ k . (3.54)( vi ) (cid:18) b P ( γ ) − ( z, k − , k ) b R ( γ ) − ( z, k − , k ) (cid:19) = T ( z, k ) − (cid:18) b P ( γ ) − ( z, k, k ) b R ( γ ) − ( z, k, k ) (cid:19) , k ≤ k , b P ( γ ) − ( z, k , k ) = ( − γ b R ( γ ) − ( z, k , k ) , k odd, − zγ ∗ b R ( γ ) − ( z, k , k ) , k even. (3.55)Next, we denote by (cid:16) P ( γ ) ± ( z,k,k ) R ( γ ) ± ( z,k,k ) (cid:17) k ∈ Z and (cid:16) Q ( γ ) ± ( z,k,k ) S ( γ ) ± ( z,k,k ) (cid:17) k ∈ Z , z ∈ C \{ } , four linearly independentsolutions of (3.21) satisfying the following initial conditions: (cid:18) P ( γ )+ ( z, k , k ) R ( γ )+ ( z, k , k ) (cid:19) = (cid:0) zγ / γ − / (cid:1) , k odd, (cid:0) γ − / γ / (cid:1) , k even, (cid:18) Q ( γ )+ ( z, k , k ) S ( γ )+ ( z, k , k ) (cid:19) = (cid:0) zγ / − γ − / (cid:1) , k odd, (cid:0) − γ − / γ / (cid:1) , k even. (3.56) (cid:18) P ( γ ) − ( z, k , k ) R ( γ ) − ( z, k , k ) (cid:19) = (cid:0) γ / − γ − / (cid:1) , k odd, (cid:0) − zγ − / γ / (cid:1) , k even, (cid:18) Q ( γ ) − ( z, k , k ) S ( γ ) − ( z, k , k ) (cid:19) = (cid:0) γ / γ − / (cid:1) , k odd, (cid:0) zγ − / γ / (cid:1) , k even. (3.57)Then, P ( γ ) ± ( z, k, k ), Q ( γ ) ± ( z, k, k ), R ( γ ) ± ( z, k, k ), and S ( γ ) ± ( z, k, k ), k, k ∈ Z , are C m × m -valued Laurent polynomials in z . INIMAL RANK DECOUPLING OF CMV OPERATORS 19
Lemma 3.9.
Let z ∈ C \{ } , k ∈ Z , and let γ j ∈ C m × m , j = 1 , , be unitary. With C = ( γ − / γ / + γ / γ − / ) and D = ( γ − / γ / − γ / γ − / ) , then, (cid:18) Q ( γ ) ± ( z, · , k ) S ( γ ) ± ( z, · , k ) (cid:19) = (cid:18) Q ( γ ) ± ( z, · , k ) S ( γ ) ± ( z, · , k ) (cid:19) C + (cid:18) P ( γ ) ± ( z, · , k ) R ( γ ) ± ( z, · , k ) (cid:19) D (3.58) (cid:18) P ( γ ) ± ( z, · , k ) R ( γ ) ± ( z, · , k ) (cid:19) = (cid:18) Q ( γ ) ± ( z, · , k ) S ( γ ) ± ( z, · , k ) (cid:19) D + (cid:18) P ( γ ) ± ( z, · , k ) R ( γ ) ± ( z, · , k ) (cid:19) C . (3.59) With C = (2 z k (mod 2) ) − ( γ − / γ / − zγ / γ − / ) , D = (2 z k (mod 2) ) − ( γ − / γ / + zγ / γ − / ) , then (cid:18) Q ( γ ) − ( z, · , k ) S ( γ ) − ( z, · , k ) (cid:19) = (cid:18) Q ( γ )+ ( z, · , k ) S ( γ )+ ( z, · , k ) (cid:19) C + (cid:18) P ( γ )+ ( z, · , k ) R ( γ )+ ( z, · , k ) (cid:19) D , (3.60) (cid:18) P ( γ ) − ( z, · , k ) R ( γ ) − ( z, · , k ) (cid:19) = (cid:18) Q ( γ )+ ( z, · , k ) S ( γ )+ ( z, · , k ) (cid:19) D + (cid:18) P ( γ )+ ( z, · , k ) R ( γ )+ ( z, · , k ) (cid:19) C . (3.61) With C = ( γ − / e ρ − k α k γ − / − γ / ρ − k α ∗ k γ / ) + ( γ − / e ρ − k γ / − γ / ρ − k γ − / ) and D = ( γ − / e ρ − k α k γ − / + γ / ρ − k α ∗ k γ / ) + ( γ − / e ρ − k γ / + γ / ρ − k γ − / ) , then (cid:18) Q ( γ ) − ( z, · , k − S ( γ ) − ( z, · , k − (cid:19) = (cid:18) Q ( γ )+ ( z, · , k ) S ( γ )+ ( z, · , k ) (cid:19) C + (cid:18) P ( γ )+ ( z, · , k ) R ( γ )+ ( z, · , k ) (cid:19) D . (3.62) With C = − ( γ − / e ρ − k α k γ − / + γ / ρ − k α ∗ k γ / ) + ( γ − / e ρ − k γ / + γ / ρ − k γ − / ) and D = − ( γ − / e ρ − k α k γ − / − γ / ρ − k α ∗ k γ / ) + ( γ − / e ρ − k γ / − γ / ρ − k γ − / ) , (cid:18) P ( γ ) − ( z, · , k − R ( γ ) − ( z, · , k − (cid:19) = (cid:18) Q ( γ )+ ( z, · , k ) S ( γ )+ ( z, · , k ) (cid:19) C + (cid:18) P ( γ )+ ( z, · , k ) R ( γ )+ ( z, · , k ) (cid:19) D . (3.63) Proof.
Since each sequence in equations (3.58)–(3.63) satisfies the recurrence relation (3.22), it isnecessary only to check equality at k = k . Substituting (3.56), (3.57) into (3.58)–(3.61) suffices toverify the identities. In addition to use of (3.56), (3.57), application of the transfer matrix T ( z, k )in (3.23) to the left sides of (3.62), (3.63) suffices in the verification of the later two identities. (cid:3) Next, we introduce half-lattice Weyl-Titchmarsh m-functions associated with the half-lattice CMVoperators, U ( γ ) ± ,k , described in (3.26).Let ∆ k = { ∆ k ( ℓ ) } ℓ ∈ Z ∈ s( Z ) m × m , k ∈ Z , denote the sequences of m × m matrices defined by(∆ k )( ℓ ) = ( I m , ℓ = k, , ℓ = k, k, ℓ ∈ Z . (3.64)Then using right-multiplication by m × m matrices on s( Z ) m × m defined in Remark 3.1, we get theidentity (∆ k X )( ℓ ) = ( X, ℓ = k, , ℓ = k, X ∈ C m × m , (3.65)and hence consider ∆ k as a map ∆ k : C m × m → s( Z ) m × m . In addition, we introduce the map∆ ∗ k : s( Z ) m × m → C m × m , k ∈ Z , defined by∆ ∗ k Φ = Φ( k ) , where Φ = { Φ( k ) } k ∈ Z ∈ s( Z ) m × m . (3.66) Similarly, one introduces the corresponding maps with Z replaced by [ k , ±∞ ) ∩ Z , k ∈ Z , which,for notational brevity, we will also denote by ∆ k and ∆ ∗ k , respectively.For k ∈ Z , and for a unitary γ ∈ C m × m , let d Ω ( γ ) ± ( · , k ) denote the C m × m -valued measures on ∂ D defined by d Ω ( γ ) ± ( ζ, k ) = d (∆ ∗ k E U ( γ ) ± ,k ( ζ )∆ k ) , ζ ∈ ∂ D , (3.67)where E U ( γ ) ± ,k ( · ) denotes the family of spectral projections for the half-lattice unitary operators U ( γ ) ± ,k , U ( γ ) ± ,k = I ∂ D dE U ( γ ) ± ,k ( ζ ) ζ. (3.68)Then, the half-lattice Weyl-Titchmarsh m-functions, m ( γ ) ± ( z, k ), are defined by m ( γ ) ± ( z, k ) = ± ∆ ∗ k ( U ( γ ) ± ,k + zI )( U ( γ ) ± ,k − zI ) − ∆ k (3.69)= ± I ∂ D d Ω ( γ ) ± ( ζ, k ) ζ + zζ − z , z ∈ C \ ∂ D , (3.70)with I ∂ D d Ω ( γ ) ± ( ζ, k ) = I m . (3.71)As defined, we note that m ( γ ) ± ( z, k ) are matrix-valued Caratheodory functions: see Appendix A fordefinition and further properties.Proof of the next result in the special case when γ = I m can be found in [19, Lemma 2.13,Theorem 2.17]. The proof in the case for general unitary γ ∈ C m × m follows the same lines and isomitted here for the sake of brevity. Theorem 3.10.
Let k ∈ Z . Then, for each unitary γ ∈ C m × m , for P ( γ ) ± ,k , Q ( γ ) ± ,k , R ( γ ) ± ,k , S ( γ ) ± ,k defined in (3.56) and (3.57) , and for m ( γ ) ± ( z, k ) defined in (3.69) , the following relations hold for z ∈ C \ ( ∂ D ∪ { } ) : b U ( γ ) ± ( z, · , k ) = Q ( γ ) ± ( z, · , k ) + P ( γ ) ± ( z, · , k ) m ( γ ) ± ( z, k ) ∈ ℓ ([ k , ±∞ ) ∩ Z ) m × m , (3.72) b V ( γ ) ± ( z, · , k ) = S ( γ ) ± ( z, · , k ) + R ( γ ) ± ( z, · , k ) m ( γ ) ± ( z, k ) ∈ ℓ ([ k , ±∞ ) ∩ Z ) m × m . (3.73) Moreover, there exist unique C m × m -valued functions M ( γ ) ± ( · , k ) such that for all z ∈ C \ ( ∂ D ∪ { } ) , U ( γ ) ± ( z, · , k ) = Q ( γ )+ ( z, · , k ) + P ( γ )+ ( z, · , k ) M ( γ ) ± ( z, k ) ∈ ℓ ([ k , ±∞ ) ∩ Z ) m × m , (3.74) V ( γ ) ± ( z, · , k ) = S ( γ )+ ( z, · , k ) + R ( γ )+ ( z, · , k ) M ( γ ) ± ( z, k ) ∈ ℓ ([ k , ±∞ ) ∩ Z ) m × m . (3.75) Remark . Within the proof of Theorem 3.10, one observes that the sequences, (cid:18) U ( γ ) ± ( z, k, k ) V ( γ ) ± ( z, k, k ) (cid:19) k ∈ Z , (3.76)defined by (3.74) and (3.75), are unique up to right-multiplication by constant m × m matrices.Hence, we shall call U ( γ ) ± ( z, · , k ) the Weyl–Titchmarsh solutions of U . Note also that m ( γ ) ± ( z, k ), aswell as M ( γ ) ± ( z, k ), are said to be half-lattice Weyl–Titchmarsh m -functions associated with U ( γ ) ± ,k .(See also [69] for a comparison of various alternative notions of Weyl–Titchmarsh m -functions for U ( γ )+ ,k with scalar-valued Verblunsky coefficients.) INIMAL RANK DECOUPLING OF CMV OPERATORS 21
For fixed k ∈ Z , z ∈ C \ ( ∂ D ∪ { } ), and unitary γ ∈ C m × m , using (3.70) and Theorem 3.10, weobtain, M ( γ )+ ( z, k ) = m ( γ )+ ( z, k ) , M ( γ )+ (0 , k ) = I m . (3.77)In particular, by (3.77) and the uniqueness up to right multiplication by constant m × m matricesnoted in Remark 3.11, note for (3.72)–(3.75) that b U ( γ )+ ( z, k, k ) = U ( γ )+ ( z, k, k ) , b V ( γ )+ ( z, k, k ) = V ( γ )+ ( z, k, k ) . (3.78)Following the line of reasoning presented for the special case when γ = I m found in [19], one usesthe equations in Lemma 3.9, together with Theorem 3.10, to obtain the following equations where C , D , and C , D are defined in (3.62) and (3.63) respectively, with γ = γ = γ . M ( γ ) − ( z, k ) = (cid:2) D + D m ( γ ) − ( z, k − (cid:3)(cid:2) C + C m ( γ ) − ( z, k − (cid:3) − (3.79)= (cid:2) m ( γ ) − ( z, k ) + I m + z ( m ( γ ) − ( z, k ) − I m ) (cid:3) (3.80) × (cid:2) m ( γ ) − ( z, k ) + I m − z ( m ( γ ) − ( z, k ) − I m ) (cid:3) − , z ∈ C \ ( ∂ D ∪ { } ) ,M ( γ ) − (0 , k ) = [ γ − / e ρ − k γ − / + γ / ρ − k γ / ][ γ − / e ρ − k γ − / − γ / ρ − k γ / ] − , (3.81) m ( γ ) − ( z, k ) = (cid:2) z ( M ( γ ) − ( z, k ) + I m ) − ( M ( γ ) − ( z, k ) − I m ) (cid:3) (3.82) × (cid:2) z ( M ( γ ) − ( z, k ) + I m ) + ( M ( γ ) − ( z, k ) − I m ) (cid:3) − , z ∈ C \ ( ∂ D ∪ { } ) . By their relations to the Caratheodory functions m ( γ ) ± ( z, k ) given in (3.77) and (3.79), we see that M ( γ ) ± ( z, k ) are also matrix-valued Caratheodory functions.Next, we introduce the C m × m -valued Schur functions Φ ( γ ) ± ( · , k ), k ∈ Z , byΦ ( γ ) ± ( z, k ) = (cid:2) M ( γ ) ± ( z, k ) − I m (cid:3)(cid:2) M ( γ ) ± ( z, k ) + I m (cid:3) − , z ∈ C \ ∂ D . (3.83)Then, by (3.82) and (3.83), one verifies that M ( γ ) ± ( z, k ) = (cid:2) I m − Φ ( γ ) ± ( z, k ) (cid:3) − (cid:2) I m + Φ ( γ ) ± ( z, k ) (cid:3) , z ∈ C \ ∂ D , (3.84) m ( γ ) − ( z, k ) = (cid:2) zI m + Φ ( γ ) − ( z, k ) (cid:3) − (cid:2) zI m − Φ ( γ ) − ( z, k ) (cid:3) , z ∈ C \ ∂ D . (3.85)Moreover, by (3.56), (3.83), and Theorem 3.10, it follows, as in [19, Lemma 2.18], that for k ∈ Z , z ∈ C \ ∂ D , Φ ( γ ) ± ( z, k ) = ( zγ / V ( γ ) ± ( z, k, k ) U ( γ ) ± ( z, k, k ) − γ / , k odd, γ / U ( γ ) ± ( z, k, k ) V ( γ ) ± ( z, k, k ) − γ / , k even, (3.86)where U ( γ ) ± ( z, k, k ) and V ( γ ) ± ( z, k, k ) are sequences defined in (3.74) and (3.75). Since the Weyl-Titchmarsh solutions defined in (3.76) are unique up to right multiplication by a constant complex m × m matirx, (3.86) implies that γ − / Φ ( γ ) ± ( · , k ) γ − / is γ -independent, and hence for unitary γ , γ ∈ C m × m , and k ∈ Z , thatΦ ( γ ) ± ( · , k ) = γ / γ − / Φ ( γ ) ± ( · , k ) γ − / γ / , (3.87) M ( γ ) ± ( · , k ) = (cid:2) ( γ − / γ / + γ / γ − / ) M ( γ ) ± ( · , k ) + ( γ − / γ / − γ / γ − / ) (cid:3) × (cid:2) ( γ − / γ / − γ / γ − / ) M ( γ ) ± ( · , k ) + ( γ − / γ / + γ / γ − / ) (cid:3) − . (3.88)Full and half-lattice resolvent operators lie at the heart of our analysis of the Weyl-Titchmarshtheory for full and half-lattice CMV operators; in particular, as a tool in obtaining our Borg-Marchenko-type uniqueness results in [19] for CMV operators with matrix-valued coefficients. Hence,we conclude with a discussion of resolvent operators for a general unitary γ ∈ C m × m . First, we note the utility of the identities contained in the following lemma. This lemma wasproven in [19, Lemma 3.2] for matrix-Laurent polynomial solutions of (3.22) defined by (3.56) in thespecial case when γ = I m . The identities listed below were central to the derivation of the full-latticeresolvent operator in [19, Lemma 3.3]. Lemma 3.12.
Let k, k ∈ Z and z ∈ C \{ } . Then, for a unitary γ ∈ C m × m , the following identitieshold for the matrix-Laurent polynomial solutions of (3.22) defined in (3.56) and (3.57) : P ( γ ) ± ( z, k, k ) Q ( γ ) ± (1 /z, k, k ) ∗ + Q ( γ ) ± ( z, k, k ) P ( γ ) ± (1 /z, k, k ) ∗ = 2( − k +1 I m , (3.89) R ( γ ) ± ( z, k, k ) S ( γ ) ± (1 /z, k, k ) ∗ + S ( γ ) ± ( z, k, k ) R ( γ ) ± (1 /z, k, k ) ∗ = 2( − k I m , (3.90) P ( γ ) ± ( z, k, k ) S ( γ ) ± (1 /z, k, k ) ∗ + Q ( γ ) ± ( z, k, k ) R ( γ ) ± (1 /z, k, k ) ∗ = 0 , (3.91) R ( γ ) ± ( z, k, k ) Q ( γ ) ± (1 /z, k, k ) ∗ + S ( γ ) ± ( z, k, k ) P ( γ ) ± (1 /z, k, k ) ∗ = 0 . (3.92) Proof.
For the case k = k , each of the equations (3.89)–(3.92) follows from (3.56) and (3.57).The induction argument described in [19, Lemma 3.2] then suffices to establish (3.89)–(3.92) when k = k . As already noted, the proof in [19, Lemma 3.2] treats the special case when γ = I m forsolutions defined by (3.56). In all cases under consideration, the proof involves a number of cases allfollowing a similar pattern. We outline one of these cases for a solution of (3.22) defined in (3.57).Suppose equations (3.89)–(3.92) hold when k ∈ Z is even. Then utilizing (3.22) together with(3.8) and (3.9), one computes P ( γ ) − ( z, k + 1 , k ) Q ( γ ) − (1 /z, k + 1 , k ) ∗ + Q ( γ ) − ( z, k + 1 , k ) P ( γ ) − (1 /z, k + 1 , k ) ∗ = e ρ − k +1 α k +1 (cid:2) P ( γ ) − ( z, k, k ) Q ( γ ) − (1 /z, k, k ) ∗ + Q ( γ ) − ( z, k, k ) P ( γ ) − (1 /z, k, k ) ∗ (cid:3) α ∗ k +1 e ρ − k +1 + e ρ − k +1 (cid:2) R ( γ ) − ( z, k, k ) S ( γ ) − (1 /z, k, k ) ∗ + S ( γ ) − ( z, k, k ) R ( γ ) − (1 /z, k, k ) ∗ (cid:3)e ρ − k +1 + z e ρ − k +1 (cid:2) R ( γ ) − ( z, k, k ) Q ( γ ) − (1 /z, k, k ) ∗ + S ( γ ) − ( z, k, k ) P ( γ ) − (1 /z, k, k ) ∗ (cid:3) α ∗ k e ρ − k +1 + e ρ − k +1 α k (cid:2) P ( γ ) − ( z, k, k ) S ( γ ) − (1 /z, k, k ) ∗ + Q ( γ ) − ( z, k, k ) R ( γ ) − (1 /z, k, k ) ∗ (cid:3)e ρ − k +1 z − = 2( − k +1 (cid:2)e ρ − k +1 α k +1 α ∗ k +1 e ρ − k +1 − e ρ − k +1 (cid:3) = 2( − ( k +1)+1 I m . (3.93)Similarly, one checks all remaining cases at the point k + 1. Then by inverting the matrix T ( z, k )and utilizing (3.22) in the form (cid:18) P − ( z, k − , k ) R − ( z, k − , k ) (cid:19) = T ( z, k ) − (cid:18) P − ( z, k, k ) R − ( z, k, k ) (cid:19) , (3.94)where T ( z, k ) − = − ρ − k α ∗ k zρ − k z − e ρ − k − e ρ − k α k ! k odd, − e ρ − k α k e ρ − k ρ − k − ρ − k α ∗ k ! k even, (3.95)one verifies the equations (3.89)–(3.92) at the point k −
1. Similarly, one verifies (3.89)–(3.92) at thepoints k + 1 and k − k ∈ Z is odd. (cid:3) The next lemma introduces the half-lattice resolvent operators for U ( γ ) ± ,k ; this appears to be anew result: INIMAL RANK DECOUPLING OF CMV OPERATORS 23
Lemma 3.13.
Let z ∈ C \ ( ∂ D ∪ { } ) and fix k ∈ Z . Then, for a unitary γ ∈ C m × m , the resolvent ( U ( γ ) ± ,k − zI ) − for the unitary CMV operator U ( γ ) ± ,k on ℓ ([ k , ±∞ ) ∩ Z ) m is given in terms of itsmatrix representation in the standard basis of ℓ ([ k , ±∞ ) ∩ Z ) m by ( U ( γ )+ ,k − zI ) − = 12 z − P ( γ )+ ( z, k, k ) b U ( γ )+ (1 / ¯ z, k ′ , k ) ∗ ,k < k ′ and k = k ′ odd, b U ( γ )+ ( z, k, k ) P ( γ )+ (1 / ¯ z, k ′ , k ) ∗ ,k > k ′ and k = k ′ even, k, k ′ ∈ [ k , ∞ ) ∩ Z , (3.96) where P ( γ )+ ( z, k, k ) is defined in (3.56) , and b U ( γ )+ ( z, k, k ) is defined in (3.72) ; ( U ( γ ) − ,k − zI ) − = 12 z − b U ( γ ) − ( z, k, k ) P ( γ ) − (1 / ¯ z, k ′ , k ) ∗ ,k < k ′ and k = k ′ odd,P ( γ ) − ( z, k, k ) b U ( γ ) − (1 / ¯ z, k ′ , k ) ∗ ,k > k ′ and k = k ′ even, k, k ′ ∈ ( −∞ , k ] ∩ Z , (3.97) where P ( γ ) − ( z, k, k ) is defined in (3.57) , and b U ( γ ) − ( z, k, k ) is defined in (3.72) .Proof. We begin by noting that the following equations hold for k ∈ Z : R ( γ ) ± ( z, k, k ) b U ( γ ) ± (1 / ¯ z, k, k ) ∗ + b V ( γ ) ± ( z, k, k ) P ( γ ) ± (1 / ¯ z, k, k ) ∗ = 0 , (3.98) P ( γ ) ± ( z, k, k ) b U ( γ ) ± (1 / ¯ z, k, k ) ∗ + b U ( γ ) ± ( z, k, k ) P ( γ ) ± (1 / ¯ z, k, k ) ∗ = 2( − k +1 I m . (3.99)These equations are a consequence of (3.72), (3.73), (3.89), (3.92), and the fact that m ( γ ) ± ( z, k ) arematrix-valued Caratheodory functions and hence satisfy the property given in (A.9).For k, k ′ ∈ [ k , ∞ ) ∩ Z , let G ( γ )+ ( z, k, k ′ , k ) be defined by G ( γ )+ ( z, k, k ′ , k ) = ( − P ( γ )+ ( z, k, k ) b U ( γ )+ (1 / ¯ z, k ′ , k ) ∗ , k < k ′ and k = k ′ odd, b U ( γ )+ ( z, k, k ) P ( γ )+ (1 / ¯ z, k ′ , k ) ∗ , k > k ′ and k = k ′ even. (3.100)Then, (3.96) is equivalent to showing that (cid:0) U ( γ )+ ,k − zI (cid:1) G ( γ )+ ( z, · , k ′ , k ) = 2 z ∆ k ′ , k ′ ∈ [ k , ∞ ) ∩ Z . (3.101)Assume that k ′ ∈ [ k , ∞ ) ∩ Z is odd. Then, for ℓ ∈ ([ k , ∞ ) ∩ Z ) \{ k ′ , k ′ + 1 } , note that (cid:0)(cid:0) U ( γ )+ ,k − zI (cid:1) G ( γ )+ ( z, · , k ′ , k ) (cid:1) ( ℓ ) = (cid:0)(cid:0) V ( γ )+ ,k W ( γ )+ ,k − zI (cid:1) G ( γ )+ ( z, · , k ′ , k ) (cid:1) ( ℓ ) = 0 . (3.102) Next, by (3.98), (3.99), note that (cid:0)(cid:0) U ( γ )+ ,k − zI (cid:1) G ( γ )+ ( z, · , k ′ , k ) (cid:1) ( k ′ ) (cid:0)(cid:0) U ( γ )+ ,k − zI (cid:1) G ( γ )+ ( z, · , k ′ , k ) (cid:1) ( k ′ + 1) ! = (cid:0)(cid:0) V ( γ )+ ,k W ( γ )+ ,k − zI (cid:1) G ( γ )+ ( z, · , k ′ , k ) (cid:1) ( k ′ ) (cid:0)(cid:0) V ( γ )+ ,k W ( γ )+ ,k − zI (cid:1) G ( γ )+ ( z, · , k ′ , k ) (cid:1) ( k ′ + 1) ! = Θ( k ′ + 1) − zR ( γ )+ ( z, k ′ , k ) b U ( γ )+ (1 / ¯ z, k ′ , k ) ∗ z b V ( γ )+ ( z, k ′ + 1 , k ) P ( γ )+ (1 / ¯ z, k ′ , k ) ∗ ! − z G ( γ )+ ( z, k ′ , k ′ , k )) G ( γ )+ ( z, k ′ + 1 , k ′ , k )) ! = Θ( k ′ + 1) z b V ( γ )+ ( z, k ′ , k ) P ( γ )+ (1 / ¯ z, k ′ , k ) ∗ z b V ( γ )+ ( z, k ′ + 1 , k ) P ( γ )+ (1 / ¯ z, k ′ , k ) ∗ ! − z G ( γ )+ ( z, k ′ , k ′ , k )) G ( γ )+ ( z, k ′ + 1 , k ′ , k )) ! = z (cid:18) b U ( γ )+ ( z, k, k ) P ( γ )+ (1 / ¯ z, k, k ) ∗ + P ( γ )+ ( z, k, k ) b U ( γ )+ (1 / ¯ z, k, k ) ∗ (cid:19) = (cid:18) z ( − k ′ +1 I m (cid:19) = (cid:18) zI m (cid:19) . (3.103)Hence, when k ′ ∈ [ k , ∞ ) ∩ Z is odd, (3.101) is a consequence of (3.102) and (3.103).Assume that k ′ ∈ [ k , ∞ ) ∩ Z is even. Then, for ℓ ∈ ([ k , ∞ ) ∩ Z ) \{ k ′ − , k ′ } , note that (3.102)holds. Again, by (3.98) and (3.99), note that (cid:0)(cid:0) U ( γ )+ ,k − zI (cid:1) G ( γ )+ ( z, · , k ′ , k ) (cid:1) ( k ′ − (cid:0)(cid:0) U ( γ )+ ,k − zI (cid:1) G ( γ )+ ( z, · , k ′ , k ) (cid:1) ( k ′ ) ! = (cid:0)(cid:0) V ( γ )+ ,k W ( γ )+ ,k − zI (cid:1) G ( γ )+ ( z, · , k ′ , k ) (cid:1) ( k ′ − (cid:0)(cid:0) V ( γ )+ ,k W ( γ )+ ,k − zI (cid:1) G ( γ )+ ( z, · , k ′ , k ) (cid:1) ( k ′ ) ! = Θ( k ′ ) − zR ( γ )+ ( z, k ′ − , k ) b U ( γ )+ (1 / ¯ z, k ′ − , k ) ∗ z b V ( γ )+ ( z, k ′ , k ) P ( γ )+ (1 / ¯ z, k ′ , k ) ∗ ! − z G ( γ )+ ( z, k ′ − , k ′ , k )) G ( γ )+ ( z, k ′ , k ′ , k )) ! = Θ( k ′ ) − zR ( γ )+ ( z, k ′ − , k ) b U ( γ )+ (1 / ¯ z, k ′ − , k ) ∗ − zR ( γ )+ ( z, k ′ , k ) b U ( γ )+ (1 / ¯ z, k ′ , k ) ∗ ! − z G ( γ )+ ( z, k ′ − , k ′ , k )) G ( γ )+ ( z, k ′ , k ′ , k )) ! = − z (cid:18) P ( γ )+ ( z, k, k ) b U ( γ )+ (1 / ¯ z, k, k ) ∗ + b U ( γ )+ ( z, k, k ) P ( γ )+ (1 / ¯ z, k, k ) ∗ (cid:19) = (cid:18) z ( − k ′ +2 I m (cid:19) = (cid:18) zI m (cid:19) . (3.104)Hence, when k ′ ∈ [ k , ∞ ) ∩ Z is even, (3.101) is a consequence of (3.102) and (3.104).The proof of (3.97) is omitted here for brevity, but follows a line of reasoning similar that justcompleted for the proof of (3.96) by using (3.72), (3.73), (3.98), and (3.99). (cid:3) Before stating our final result for the full-lattice resolvent of U , let us recall the definition of, andsome facts about, the matrix-valued Wronskian, defined in [19], for two C m × m -valued sequences U j ( z, · ), j = 1 ,
2. First, the Wronskian is defined for k ∈ Z , z ∈ C \{ } , by W ( U (1 /z, k ) , U ( z, k ))= ( − k +1 (cid:2) U (1 /z, k ) ∗ U ( z, k ) − ( V ∗ U (1 /z, · ))( k ) ∗ ( V ∗ U ( z, · ))( k ) (cid:3) , (3.105) INIMAL RANK DECOUPLING OF CMV OPERATORS 25 where V is defined in (3.15). It is shown in [19, Lemma 3.1] when U U j ( z, · ) = zU j ( z, · ), and hence V ∗ U j ( z, · ) = V j ( z, · ), j = 1 ,
2, where U is viewed as a difference expression rather than as an operatoracting on ℓ ( Z ) m × m , that the Wronskian of U j ( z, · ), j = 1 ,
2, is independent of k ∈ Z . Moreover,for P ( γ )+ ( z, · , k ) and Q ( γ )+ ( z, · , k ) defined in (3.56), and for U ( γ ) ± ( z, · , k ) defined in (3.74), with k, k ∈ Z , z ∈ C \{ } , as a consequence of (3.56), (3.57), (3.74), (3.75), and property (A.9), we seethat W (cid:0) P ( γ )+ (1 /z, k, k ) , Q ( γ )+ ( z, k, k ) (cid:1) = I m , (3.106) W (cid:0) U ( γ )+ (1 /z, k, k ) , U ( γ ) − ( z, k, k ) (cid:1) = M ( γ ) − ( z, k ) − M ( γ )+ ( z, k ) . (3.107)For notational simplicity, we abbreviate the Wronskian of U ( γ )+ and U ( γ ) − by W ( γ ) ( z, k ) = − W (cid:0) U ( γ )+ (1 /z, k, k ) , U ( γ ) − ( z, k, k ) (cid:1) . (3.108)Then, using (3.77), (3.81), and (3.107), one analytically continues W ( γ ) ( z, k ) to z = 0 and obtains W ( γ ) ( z, k ) = M ( γ )+ ( z, k ) − M ( γ ) − ( z, k ) , k ∈ Z , z ∈ C . (3.109)Moreover, one verifies the following symmetry property of the Wronskian W ( γ ) ( z, k ), for k ∈ Z , z ∈ C M ( γ )+ ( z, k ) W ( γ ) ( z, k ) − M ( γ ) − ( z, k ) = M ( γ ) − ( z, k ) W ( γ ) ( z, k ) − M ( γ )+ ( z, k ) . (3.110)Then, using (3.74), (3.75), (3.89), (3.92), (3.107), (3.109), and following the steps in the proof for[19, Lemma 3.2] for the special case when γ = I m , we find that U ( γ )+ ( z, k, k ) W ( γ ) ( z, k ) − U ( γ ) − (1 /z, k, k ) ∗ − U ( γ ) − ( z, k, k ) W ( γ ) ( z, k ) − U ( γ )+ (1 /z, k, k ) ∗ = 2( − k +1 I m , (3.111) V ( γ )+ ( z, k, k ) W ( γ ) ( z, k ) − U ( γ ) − (1 /z, k, k ) ∗ − V ( γ ) − ( z, k, k ) W ( γ ) ( z, k ) − U ( γ )+ (1 /z, k, k ) ∗ = 0 . (3.112)Then, using (3.111) and (3.112), and following the steps in the proof for [19, Lemma 3.3] for thespecial case when γ = I m , we obtain the next result for the resolvent of the full-lattice operator U . Lemma 3.14.
Let z ∈ C \ ( ∂ D ∪ { } ) , fix k ∈ Z , and let γ ∈ C m × m be unitary. Then the resolvent ( U − zI ) − of the unitary CMV operator U on ℓ ( Z ) m is given, for k, k ′ ∈ Z , in terms of its matrixrepresentation in the standard basis of ℓ ( Z ) m by ( U − zI ) − ( k, k ′ ) = 12 z U ( γ ) − ( z, k, k ) W ( γ ) ( z, k ) − U ( γ )+ (1 /z, k ′ , k ) ∗ ,k < k ′ or k = k ′ odd ,U ( γ )+ ( z, k, k ) W ( γ ) ( z, k ) − U ( γ ) − (1 /z, k ′ , k ) ∗ ,k > k ′ or k = k ′ even , (3.113) Moreover, since ∈ C \ σ ( U ) , (3.113) analytically extends to z = 0 . Appendix A. Basic Facts on Caratheodory and Schur Functions
In this appendix we summarize a few basic facts on matrix-valued Caratheodory and Schurfunctions used throughout this manuscript. (For the analogous case of matrix-valued Herglotzfunctions we refer to [48] and the extensive list of references therein.)We denote by D and ∂ D the open unit disk and the counterclockwise oriented unit circle in thecomplex plane C , D = { z ∈ C | | z | < } , ∂ D = { ζ ∈ C | | ζ | = 1 } . (A.1) Moreover, we denote as usual Re( A ) = ( A + A ∗ ) / A ) = ( A − A ∗ ) / (2 i ) for square matrices A with complex-valued entries. Definition A.1.
Let m ∈ N and F ± , Φ + , and Φ − − be m × m matrix-valued analytic functions in D .( i ) F + is called a Caratheodory matrix if Re( F + ( z )) ≥ z ∈ D and F − is called an anti-Caratheodory matrix if − F − is a Caratheodory matrix.( ii ) Φ + is called a Schur matrix if k Φ + ( z ) k C m × m ≤
1, for all z ∈ D . Φ − is called an anti-Schur matrix if Φ − − is a Schur matrix. Theorem A.2.
Let F be an m × m Caratheodory matrix, m ∈ N . Then F admits the Herglotzrepresentation F ( z ) = iC + I ∂ D d Ω( ζ ) ζ + zζ − z , z ∈ D , (A.2) C = Im( F (0)) , I ∂ D d Ω( ζ ) = Re( F (0)) , (A.3) where d Ω denotes a nonnegative m × m matrix-valued measure on ∂ D . The measure d Ω can bereconstructed from F by the formula Ω (cid:0) Arc (cid:0)(cid:0) e iθ , e iθ (cid:3)(cid:1)(cid:1) = lim δ ↓ lim r ↑ π I θ + δθ + δ dθ Re (cid:0) F (cid:0) rζ (cid:1)(cid:1) , (A.4) where Arc (cid:0)(cid:0) e iθ , e iθ (cid:3)(cid:1) = (cid:8) ζ ∈ ∂ D | θ < θ ≤ θ (cid:9) , θ ∈ [0 , π ) , θ < θ ≤ θ + 2 π. (A.5) Conversely, the right-hand side of equation (A.2) with C = C ∗ and d Ω a finite nonnegative m × m matrix-valued measure on ∂ D defines a Caratheodory matrix. We note that additive nonnegative m × m matrices on the right-hand side of (A.2) can be absorbedinto the measure d Ω since I ∂ D dµ ( ζ ) ζ + zζ − z = 1 , z ∈ D , (A.6)where dµ ( ζ ) = dθ π , ζ = e iθ , θ ∈ [0 , π ) (A.7)denotes the normalized Lebesgue measure on the unit circle ∂ D .Given a Caratheodory (resp., anti-Caratheodory) matrix F + (resp. F − ) defined in D as in (A.2),one extends F ± to all of C \ ∂ D by F ± ( z ) = iC ± ± I ∂ D d Ω ± ( ζ ) ζ + zζ − z , z ∈ C \ ∂ D , C ± = C ∗± . (A.8)In particular, F ± ( z ) = − F ± (1 /z ) ∗ , z ∈ C \ D . (A.9)Of course, this continuation of F ± | D to C \ D , in general, is not an analytic continuation of F ± | D .Next, given the functions F ± defined in C \ ∂ D as in (A.8), we introduce the functions Φ ± byΦ ± ( z ) = [ F ± ( z ) − I m ][ F ± ( z ) + I m ] − , z ∈ C \ ∂ D . (A.10)We recall (cf., e.g., [78, p. 167]) that if ± Re( F ± ) ≥
0, then [ F ± ± I m ] is invertible. In particular,Φ + | D and [Φ − ] − | D are Schur matrices (resp., Φ − | D is an anti-Schur matrix). Moreover, F ± ( z ) = [ I m − Φ ± ( z )] − [ I m + Φ ± ( z )] , z ∈ C \ ∂ D . (A.11) Acknowledgments.
Fritz Gesztesy would like to thank all organizers of the 14th InternationalConference on Difference Equations and Applications (ICDEA 2008), for their kind invitation and
INIMAL RANK DECOUPLING OF CMV OPERATORS 27 the stimulating atmosphere created during the meeting. In addition, he is particularly indebted toMehmet ¨Unal for the extraordinary hospitality extended to him during his ten day stay in Istanbulin July of 2008.
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Department of Mathematics & Statistics, University of Missouri, Rolla, MO 65409, USA
E-mail address : [email protected] URL : http://web.umr.edu/~sclark/index.html Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
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