Contractive determinantal representations of stable polynomials on a matrix polyball
Anatolii Grinshpan, Dmitry S. Kaliuzhnyi-Verbovetskyi, Victor Vinnikov, Hugo J. Woerdeman
aa r X i v : . [ m a t h . F A ] M a r CONTRACTIVE DETERMINANTAL REPRESENTATIONS OF STABLEPOLYNOMIALS ON A MATRIX POLYBALL
ANATOLII GRINSHPAN, DMITRY S. KALIUZHNYI-VERBOVETSKYI, VICTOR VINNIKOV,AND HUGO J. WOERDEMAN
Abstract.
We show that an irreducible polynomial p with no zeros on the closure of amatrix unit polyball, a.k.a. a cartesian product of Cartan domains of type I, and such that p (0) = 1, admits a strictly contractive determinantal representation, i.e., p = det( I − KZ n ),where n = ( n , . . . , n k ) is a k -tuple of nonnegative integers, Z n = L kr =1 ( Z ( r ) ⊗ I n r ), Z ( r ) =[ z ( r ) ij ] are complex matrices, p is a polynomial in the matrix entries z ( r ) ij , and K is a strictlycontractive matrix. This result is obtained via a noncommutative lifting and a theorem onthe singularities of minimal noncommutative structured system realizations. Introduction
Polynomial stability arises naturally in various problems of Analysis and its applicationssuch as Electrical Engineering and Control Theory [8, 26, 7, 19, 9, 14, 23, 22]. A polynomial p ∈ C [ z , . . . , z d ] is called stable with respect to a domain D ⊆ C d , or just D -stable, if it hasno zeros in D , and strongly D -stable if it has no zeros in the domain closure D . In the casewhere d = 1 and D is the unit disk D = { z ∈ C : | z | < } , and p (0) = 1, one can write p = (1 − a z ) · · · (1 − a n z ) = det( I − Kz ) , where a i = 1 /z i , i = 1 , . . . , n , the zeros z i of p are counted according to their multiplicities, K = diag[ a , . . . , a n ], and n = deg p . It follows that the matrix K is contractive (resp.,strictly contractive), i.e., k K k ≤ k K k < k · k isthe operator (2 ,
2) norm.In the case where d = 2 and D is the unit bidisk D , it is also true that a stable (resp.,strongly stable) polynomial p has a contractive (resp., strictly contractive) determinantalrepresentation. It was shown in [10] (see also [20, 19, 21]) that every D -stable (resp.,strongly D -stable) polynomial p , with p (0) = 1, can be represented as(1.1) p = det( I − KZ n ) , where n = ( n , n ) ∈ Z is the bi-degree of p , Z n = diag[ z I n , z I n ] , and the matrix K iscontractive (resp., strictly contractive).For a higher-dimensional polydisk case, D = D d , d >
2, it is in general not true that everystable (resp., strongly stable) polynomial p , with p (0) = 1, has a determinantal representation Mathematics Subject Classification.
Key words and phrases.
Contractive determinantal representation; stable polynomial; polyball; classicalCartan domain; contractive realization; structured noncommutative multidimensional system.AG, DK-V, HW were partially supported by NSF grant DMS-0901628. DK-V and VV were partiallysupported by BSF grant 2010432. Here and in the rest of the paper we use a convention that a matrix block which involves I n i is void inthe case of n i equal to 0. n = ( n , . . . , n d ) ∈ Z d + is equal to the multi-degree of p , deg p , Z n =diag[ z I n , . . . , z d I n d ], and the matrix K is contractive (resp., strictly contractive). Such arepresentation with n ≥ deg p (in the sense that n i ≥ deg i p , i = 1 , . . . , d , where deg i p denotes the i -th partial degree of p ) has been constructed for some special classes of stablepolynomials in [12].The existence of a representation (1.1) with a contractive (resp., strictly contractive) ma-trix K provides a certificate for stability (resp., strong stability) of a polynomial p . Moreover,if merely a polynomial multiple of p has such a representation, the stability (resp., strongstability) of p is guaranteed. In a recent paper of the authors [11], the following result hasbeen obtained. Let P ∈ C ℓ × m [ z , . . . , z d ] and P ( r ) ∈ C ℓ r × m r [ z , . . . , z d ], r = 1 , . . . , k , be suchthat P = L kr =1 P ( r ) , and let(1.2) D P = { z ∈ C d : k P ( z ) k < } . Under an appropriate Archimedean condition on P , which in particular implies the bound-edness of the domain D P , for every strongly D P -stable polynomial p ∈ C [ z , . . . , z d ] thereexists a polynomial q ∈ C [ z , . . . , z d ] such that pq has a determinantal representation(1.3) pq = det( I − K P n ) , where n = ( n , . . . , n k ) ∈ Z k + , P n = L kr =1 ( P ( r ) ⊗ I n r ), and the matrix K is strictly contrac-tive. We note that special cases of the domain D P as above include the unit polydisk andthe classical Cartan domains of type I, II, and III, as well as the Cartesian products of suchdomains.In this paper, we construct strictly contractive determinantal representations for stronglystable polynomials on a Cartesian product of Cartan’s domains of type I, i.e., on a matrixunit polyball,(1.4) B ℓ × m × · · · × B ℓ k × m k = n Z = ( Z (1) , . . . , Z ( k ) ) ∈ C ℓ × m × · · · × C ℓ k × m k : k Z ( r ) k < , r = 1 , . . . , k o . In other words, in the case where D P is a unit matrix polyball, i.e., where P ( r ) = Z ( r ) , r = 1 , . . . , k , no additional polynomial factor q is needed to construct a strictly contractivedeterminantal representation (1.3), and we have p = det( I − KZ n ) , Z n = k M r =1 ( Z ( r ) ⊗ I n r )with some n = ( n , . . . , n k ) ∈ Z k + and some P kr =1 m r n r × P kr =1 ℓ r n r matrix K such that k K k <
1. Notice that the unit polydisk D d is a special case of a unit matrix polyball where k = d , and ℓ r = m r = 1 for r = 1 , . . . , k .The proof of our main theorem has two components: realization formulas for multivariablerational functions and related techniques from multidimensional system theory, and resultsfrom a theory of noncommutative rational functions.The first component was a key for constructing determinantal representations in [20, 21,19, 12, 10, 11]. We recall (see [3, Proposition 11]) that every matrix-valued rational functionthat is regular and contractive on the open unit disk D can be realized as F = D + zC ( I − zA ) − B, ith a contractive colligation matrix [ A BC D ]. In several variables, the celebrated result ofAgler [1] gives the existence of a realization of the form F ( z ) = D + CZ X ( I − AZ X ) − B, Z X = d M i =1 z i I X i , where z = ( z , . . . , z d ) ∈ D d and the colligation [ A BC D ] is a Hilbert-space unitary operator(with A acting on the orthogonal direct sum of Hilbert spaces X , . . . , X d ), for F an operator-valued function holomorphic on the unit polydisk D d whose Agler norm k F k A = sup T ∈T k F ( T ) k is at most 1. Here T is the set of d -tuples T = ( T , . . . , T d ) of commuting strict contractionson a Hilbert space. Such functions constitute the Schur–Agler class.Agler’s result was generalized to polynomially defined domains in [2, 4]. Given P ∈ C ℓ × m [ z , . . . , z d ], let D P be as in (1.2) (here we can assume that k = 1 and P = P ) and let T P be the set of d -tuples T of commuting bounded operators on a Hilbert space satisfying k P ( T ) k <
1. (The case of unit polydisk D d corresponds to P = diag[ z , . . . , z d ] and T P = T .)For T ∈ T P , the Taylor joint spectrum σ ( T ) [24] lies in D P (see [2, Lemma 1]), and thereforefor an operator-valued function F holomorphic on D P one defines F ( T ) by means of Taylor’sfunctional calculus [25] and(1.5) k F k A , P = sup T ∈T P k F ( T ) k . We say that F belongs to the operator-valued Schur–Agler class associated with P , denotedby SA P ( U , Y ), if F is holomorphic on D P , takes values in the space L ( U , Y ) of boundedlinear operators from a Hilbert space U to a Hilbert space Y , and k F k A , P ≤ F belongs to the Schur–Agler class SA P ( U , Y ) if and only if there exist a Hilbert space X and a unitary colligation (cid:20) A BC D (cid:21) : ( C m ⊗ X ) ⊕ U → ( C ℓ ⊗ X ) ⊕ Y such that(1.6) F ( z ) = D + C ( P ( z ) ⊗ I X ) (cid:16) I − A ( P ( z ) ⊗ I X ) (cid:17) − B. If the Hilbert spaces U and Y are finite-dimensional, F can be treated as a matrix-valuedfunction (relative to a pair of orthonormal bases for U and Y ). It is natural to ask whetherevery rational α × β matrix-valued function in the Schur–Agler class SA P ( C β , C α ) has arealization (1.6) with a contractive colligation matrix [ A BC D ]. This question is open for d > F is an inner (i.e., regular on D d andtaking unitary boundary values a.e. on the unit torus T d = { z = ( z , . . . , z d ) ∈ C d : | z i | =1 , i = 1 , . . . , d } ) matrix-valued Schur–Agler function on D d . In this case, the colligationmatrix for the realization (1.6) can be chosen unitary; see [18] for the scalar-valued case,and [6, Theorem 2.1] for the matrix-valued generalization. We notice here that not everyrational inner function is Schur–Agler; see [12, Example 5.1] for a counterexample. In thesecond case, one assumes that P = L kr =1 P ( r ) satisfies a certain matrix-valued Archimedean ondition and F is regular on the closed domain D P and satisfies k F k A , P <
1. Then thereexists a contractive finite-dimensional realization of F in the form(1.7) F = D + C P n ( I − A P n ) − B, P n = k M r =1 ( P ( r ) ⊗ I n r ) , with some n = ( n , . . . , n k ) ∈ Z k + [11].The second component in the proof of our main result, a theory of noncommutative ra-tional functions, is briefly summarized in Section 2. Then a version of a theorem from [17]on the singularities of a noncommutative rational matrix-valued function in terms of itsminimal realization, where the realization is in the form of a structured noncommutativemultidimensional system, i.e., the one that is associated with a unit polyball (1.4), is proved(see [5] for details on structured noncommutative multidimensional systems). As a corollary,an analogous theorem on the singularities of a commutative matrix-valued rational functionis obtained via a noncommutative lifting.In Section 3, our main theorem is proved, which establishes the existence of a strictlycontractive determinantal representation for every irreducible strongly stable polynomial ona matrix polyball. As a corollary, in the case of the unit polydisk D d , we obtain that everystrongly stable polynomial p is an eventual Agler denominator, i.e., is the denominator of arational inner function of the Schur–Agler class.2. Singularities of noncommutative rational functions and minimalstructured noncommutative multidimensional systems
We first give some necessary background on matrix-valued noncommutative rational func-tions; see [17, 16] for more details, and we also refer to [15] for a general theory of freenoncommutative functions.A matrix-valued noncommutative rational expression R over a field K is any expressionobtained from noncommuting indeterminates z , . . . , z d , and a constant 1 ∈ K by successiveelementary operations: addition, multiplication, and inversion, forming (block) matrices,and also matrix addition, multiplication, and inversion. E.g., an α × β matrix-valued non-commutative polynomial R = X w ∈G d : | w |≤ L R w z w is a matrix-valued noncommutative rational expression defined without using inversions.Here G d is the free monoid on d generators (letters) g , . . . , g d , the coefficients R w are α × β matrices over K , and for an element w = g i · · · g i N ∈ G d (a word in the alphabet g , . . . , g d )we set z w = z i · · · z i N and z ∅ = 1, where ∅ is the unit element of G d (the empty word), and | w | = N is the length of the word w , in particular |∅| = 0.For a d -tuple Z = ( Z , . . . , Z d ) of s × s matrices over K , one can evaluate R ( Z ) = R s ( Z ) = X w ∈G d R w ⊗ Z w , where Z w = Z i · · · Z i k and Z ∅ = I s . Similarly, one can evaluate R on a d -tuple X =( X , . . . , X d ) of generic s × s matrices, i.e., on a d -tuple of matrices over commuting inde-terminates ( X r ) ij , r = 1 , . . . , d , i, j = 1 , . . . , s . We then define evaluations R ( Z ) = R s ( Z )and R ( X ) = R s ( X ) whenever all the formal matrix inversions in the expression R can be eplaced by matrix inversions for matrices over K (resp., for generic matrices); this definesthe domain of regularity of R , dom s R , and the extended domain of regularity of R , edom s R ,inside the set of d -tuples of s × s matrices over K ( d -tuples of generic matrices). Then onedefines dom R = ` ∞ s =1 dom s R and edom R = ` ∞ s =1 edom s R . One hasedom s R ⊇ dom s R, edom R ⊇ dom R. Two α × β matrix-valued noncommutative rational expressions R and R are called equiv-alent if dom R ∩ dom R = ∅ and R ( Z ) = R ( Z ) for every Z ∈ dom R ∩ dom R . Anequivalence class of α × β matrix-valued noncommutative rational expressions is called an α × β matrix-valued noncommutative rational function. We write R ∈ R if a matrix-valuednoncommutative rational function R as an equivalence class of matrix-valued noncommuta-tive rational expressions contains R . We definedom s R = \ R ∈ R dom s R, dom R = ∞ a s =1 dom s R . Next, we observe that if R and R are equivalent, then their evaluations on generic matricesgive rise to the same α × β matrix-valued commutative rational function, so edom s R =edom s R for every s . Therefore, we can defineedom s R = edom s R, edom R = edom R for any R ∈ R . Clearly, we haveedom s R ⊇ dom s R , edom R ⊇ dom R . In [17], the left and right backward shift operators L j and R j , j = 1 , . . . , d , were definedfor matrix-valued noncommutative rational expressions. It was shown that if R and R are equivalent, then so are L j ( R ) and L j ( R ) (resp., R j ( R ) and R j ( R )). Therefore,these definitions can be extended to matrix-valued noncommutative rational functions. Onedefines dom L j ( R ) = dom R j ( R ) = dom R, however we have edom L j ( R ) ⊇ edom R, edom R j ( R ) ⊇ edom R, and therefore edom L j ( R ) ⊇ edom R , edom R j ( R ) ⊇ edom R . We will not need the general definitions of the left and right backward shifts here. It sufficesfor us to use the fact that every matrix-valued noncommutative rational function R whichis regular at 0, i.e., such that 0 ∈ dom R , has a formal power series expansion R ∼ X w ∈G d R w z w , whose evaluation on s × s matrices is convergent in some neighborhood of zero for each s ,and that L j R ∼ X w ∈G d R g j w z w , R j R ∼ X w ∈G d R wg j z w . The following theorem is a structured-system analogue of [17, Theorem 3.1]; for details onstructured noncommutative multidimensional systems, see [5]. We note that we are not usinghere a bipartite-graph formalism adopted in [5] for system evolutions and, as a consequence, or the definitions of controllability and observability. Instead, we use more direct block-matrix notations. The diligent reader can easily find the one-to-one correspondence betweenthe two formalisms. Theorem 2.1.
Let R be an α × β matrix-valued noncommutative rational function over afield K represented by the expression (2.1) R = D + Cz n ( I − Az n ) − B, where n = ( n , . . . , n k ) ∈ Z k + , z n = L kr =1 ( z ( r ) ⊗ I n r ) , z ( r ) = [ z ( r ) ij ] is a ℓ r × m r matrixwhose entries z ( r ) ij are noncommuting indeterminates, [ A BC D ] ∈ K ( P kr =1 m r n r + α ) × ( P kr =1 ℓ r n r + β ) isa block matrix whose blocks A, B, C have further block decompositions : A = [ A ( rr ′ ) ] r,r ′ =1 ,...,k with blocks A ( rr ′ ) ∈ K m r × ℓ r ′ ⊗ K n r × n r ′ ∼ = ( K n r × n r ′ ) m r × ℓ r ′ , so that for j = 1 , . . . , m r , i =1 , . . . , ℓ r ′ one has A ( rr ′ ) ji ∈ K n r × n r ′ ; B = col r =1 ,...,k [ B ( r ) ] with blocks B ( r ) ∈ K m r × ⊗ K n r × β ∼ =( K n r × β ) m r × , so that for j = 1 , . . . , m r one has B ( r ) j ∈ K n r × β ; C = row i =1 ,...,k [ C ( r ) ] withblocks C ( r ) ∈ K × ℓ r ⊗ K α × n r ∼ = ( K α × n r ) × ℓ r , so that for i = 1 , . . . , ℓ r one has C ( r ) i ∈ K α × n r ;and D ∈ K α × β . Assume that the realization R of R as in (2.1) is minimal, or equivalently,controllable, i.e., for each r ∈ { , . . . , k } and j ∈ { , . . . , m r } one has (2.2) X N ∈ N , γ ∈{ ,...,N } , i γ ∈{ ,...,ℓ rγ } , j γ ∈{ ,...,m rγ } range( A ( r r ) j i · · · A ( r N − r N ) j N − i N B ( r N ) j N ) = K n r , and observable, i.e., for each r ∈ { , . . . , k } and i ∈ { , . . . , ℓ r } one has (2.3) \ N ∈ N , γ ∈{ ,...,N } , i γ ∈{ ,...,ℓ rγ } , j γ ∈{ ,...,m rγ } ker( C ( r N ) i N A ( r N r N − ) j N i N − · · · A ( r r ) j i ) = { } . Then (2.4) edom R = dom R = ∞ a s =1 n Z = ( Z (1) , . . . , Z ( k ) ) ∈ ( K s × s ) ℓ × m × · · · × ( K s × s ) ℓ k × m k ∼ = ( K ℓ × m × · · · × K ℓ k × m k ) ⊗ K s × s : det( I − A ⊙ Z ) = 0 o , where A ⊙ Z ∈ K P kr =1 m r n r s × P kr =1 m r n r s is a block P kr =1 m r × P kr =1 m r matrix with blocks ( A ⊙ Z ) ( rr ′ ) ij = ℓ r ′ X κ =1 A ( rr ′ ) iκ ⊗ Z ( r ′ ) κj ∈ K n r × n r ′ ⊗ K s × s ∼ = K n r s × n r ′ s ,i = 1 , . . . , m r , j = 1 , . . . , m r ′ .Proof. It is clear that the inclusion “ ⊇ ” holds in both the equalities in (2.4).Conversely, let Z ∈ edom s R for some s ∈ N . We will show that det( I − A ⊙ Z ) = 0 . Let L = ( L ( r ) ij ) r =1 ,...,k, i =1 ,...,ℓ r , j =1 ,...,m r , R = ( R ( r ) ij ) i =1 ,...,ℓ r , j =1 ,...,m r Here, similarly to the convention we made in a footnote on the front page of the paper, we assume thata matrix block is void if the number of its rows/columns is 0. e the d -tuples of left and right backward shifts, where d = P kr =1 ℓ r m r . For a word w = g ( r ) i j · · · g ( r N ) i N j N ∈ G d , we set L w = L ( r ) i j · · · L ( r N ) i N j N , R w = R ( r ) i j · · · R ( r N ) i N j N . Then for any w = g ( r ) i j · · · g ( r N ) i N j N ∈ G d with N ≥ R w ( R ) = R w (cid:16) Cz n ( I − Az n ) − B (cid:17) = R w (cid:16) C ( I − z n A ) − z n B (cid:17) = C ( I − z n A ) − (cid:16) e ( r ) i ⊗ A ( r r ) j i · · · A ( r N − r N ) j N − i N B ( r N ) j N (cid:17) , where e ( r ) i is the (cid:16) P r − r =1 ℓ r + i (cid:17) -th standard basis vector of K P kr =1 ℓ r . Since Z ∈ edom s R ,we have Z ∈ edom s R w R = edom s R w ( R ). Therefore, the αs × βs matrix-valued rationalfunction ( C ⊗ I s )( I − X ⊙ op A ) − (cid:16) e ( r ) i ⊗ A ( r r ) j i · · · A ( r N − r N ) j N − i N B ( r N ) j N ⊗ I s (cid:17) in the commuting variables ( X ( r ) ij ) µν , r = 1 , . . . , k , i = 1 , . . . , ℓ r , j = 1 , . . . , m r , µ, ν =1 , . . . , s , is regular at X = Z , where X ⊙ op A ∈ K P kr =1 ℓ r n r s × P kr =1 ℓ r n r s is a block P kr =1 ℓ r × P kr =1 ℓ r matrix with blocks( X ⊙ op A ) ( rr ′ ) ij = m r X κ =1 A ( rr ′ ) κj ⊗ X ( r ) iκ ∈ K n r × n r ′ ⊗ K s × s ∼ = K n r s × n r ′ s . The controllability assumption implies that the αs × P kr =1 ℓ r n r s matrix-valued rational func-tion ( C ⊗ I s )( I − X ⊙ op A ) − is regular at X = Z . Therefore, the αs × P kr =1 m r n r s matrix-valued rational function( C ⊗ I s )( I − X ⊙ op A ) − ( X ⊙ op I P kr =1 m r n r ) = ( C ⊙ X )( I − A ⊙ X ) − is regular at X = Z . In other words, Z ∈ edom s R ′ where R ′ = Cz n ( I − Az n ) − is an α × P kr =1 m r n r matrix-valued noncommutative rational expression.Next, for any w = g ( r ) i j · · · g ( r n ) i N j N ∈ G d with N ≥ w ⊤ = g ( r N ) i N j N · · · g ( r ) i j . Then wehave L w ⊤ ( R ′ ) = (cid:16) ( f ( r ) j ) ⊤ ⊗ C ( r N ) i N A ( r N r N − ) j N i N − · · · A ( r r ) j i (cid:17) ( I − Az n ) − , where f ( r ) j is the (cid:16) P r − r =1 m r + j (cid:17) -th standard basis vector of K P kr =1 m r . Since Z ∈ edom s R ′ ,we have Z ∈ edom s L w ⊤ ( R ′ ). Therefore, the αs × P kr =1 m r n r s matrix-valued rational function (cid:16) ( f ( r ) j ) ⊤ ⊗ C ( r N ) i N A ( r N r N − ) j N i N − · · · A ( r r ) j i ⊗ I s (cid:17) ( I − A ⊙ X ) − in the commuting variables ( X ( r ) ij ) µν , r = 1 , . . . , k , i = 1 , . . . , ℓ r , j = 1 , . . . , m r , µ, ν =1 , . . . , s , is regular at X = Z . The observability assumption implies that the P kr =1 m r n r s × P kr =1 m r n r s matrix-valued rational function( I − A ⊙ X ) − is regular at X = Z . Then so is the rational functiondet( I − A ⊙ X ) − = (det( I − A ⊙ X )) − , .e., det( I − A ⊙ Z ) = 0, as required. (cid:3) Corollary 2.2.
The variety of singularities of an α × β matrix-valued rational function f which can be represented as a restriction R of an α × β matrix-valued noncommutativerational expression R of the form (2.1) satisfying the assumptions of Theorem 2.1 (i.e., whichis obtained from R by replacing the noncommuting indeterminates z ( r ) ij by the commutingones), is given by n Z = ( Z (1) , . . . , Z ( k ) ) ∈ K ℓ × m × · · · × K ℓ k × m k : det( I − AZ n ) = 0 o , where Z n = L kr =1 ( Z ( r ) ⊗ I n r ) .Proof. Clearly, the variety of singularities of f coincides with (cid:16) K ℓ × m × · · · × K ℓ k × m k (cid:17) \ edom R. The result then follows from Theorem 2.1. (cid:3)
We will also need to make use of the inverse of a noncommutative rational function, and ofthe fact that the minimality of a realization carries over to the corresponding realization ofthe inverse. We recall from [5, Section 4] that if R is an α × α matrix-valued noncommutativerational function over a field K represented by the noncommutative rational expression (2.1)with D invertible, then its inverse exists and has a realization(2.5) R − = D × + C × z n ( I − A × z n ) − B × , where(2.6) (cid:20) A × B × C × D × (cid:21) = (cid:20) A − BD − C BD − − D − C D − (cid:21) . Proposition 2.3.
Assume that the realization of R in (2.1) is minimal and that D is in-vertible. Then the realization of R − given via (2.5) and (2.6) is also minimal.Proof. It suffices to verify the controllability and observability for the realization of R − .Notice that the blocks of A × and B × are A × ( rr ′ ) ji = A ( rr ′ ) ji − B ( r ) j D − C ( r ′ ) i and B × ( r ) j = B ( r ) j D − ,respectively. Thus, for the controllability, we need to check that for each r ∈ { , . . . , N } and j ∈ { , . . . , m r } one has(2.7) X N ∈ N , γ ∈{ ,...,N } , i γ ∈{ ,...,ℓ rγ } , j γ ∈{ ,...,m rγ } range( A × ( r r ) j i · · · A × ( r N − r N ) j N − i N B × ( r N ) j N ) = K n r . Clearly, range B × ( r ) j = range B ( r ) j D − = range B ( r ) j for all r = 1 , . . . , k and j = 1 , . . . , m r .Next,range( A × ( r r ) j i B × ( r ) j ) + range B × ( r ) j = range (cid:16) ( A ( r r ) j i − B ( r ) j D − C ( r ) i ) B ( r ) j D − (cid:17) + range B ( r ) j D − = range( A ( r r ) j i B ( r ) j ) + range B ( r ) j . Continuing this way one sees that the left hand sides of (2.2) and (2.7) are the same, andthus (2.7) follows from (2.2). In a similar way, one shows the observability. (cid:3) . Contractive determinantal depresentations of stable polynomials on amatrix polyball
The main result of the paper is the following.
Theorem 3.1.
Let p be an irreducible polynomial in the commuting indeterminates z ( r ) ij , r = 1 , . . . , k , i = 1 , . . . , ℓ r , j = 1 , . . . , m r , with p (0) = 1 , which is strongly stable with respectto the matrix polyball B ℓ × m × · · · × B ℓ k × m k . Then there exist n = ( n , . . . , n k ) ∈ Z k + and astrict contraction K ∈ C P kr =1 m r n r × P kr =1 ℓ r n r so that (3.1) p = det( I − KZ n ) , where Z n = L kr =1 ( Z ( r ) ⊗ I n r ) and Z ( r ) = [ z ( r ) ij ] ∈ C ℓ r × m r .Proof. Since p has no zeros in the closed unit polyball B ℓ × m × · · · × B ℓ k × m k , we have that p has no zeros in ρ B ℓ × m × · · · × ρ B ℓ k × m k for some ρ > g = 1 /p is regular on ρ B ℓ × m × · · · × ρ B ℓ k × m k , and the rational function g ρ defined by g ρ ( z ) = g ( ρz ) is regular on B ℓ × m × · · · × B ℓ k × m k . By [11, Lemma 3.3], k g ρ k A ,Z < ∞ , where the corresponding Agler norm k · k A ,Z = k · k A ,P is defined as in (1.5)with P = Z = L kr =1 Z ( r ) . Thus we can find a constant c > k cg ρ k A ,Z <
1. By[11, Theorem 3.4] applied to F = cg ρ , we obtain a n = ( n , . . . , n k ) ∈ Z k + and a contractivecolligation matrix [ A BC D ] of size ( P kr =1 m r n r + 1) × ( P kr =1 ℓ r n r + 1) such that cg ρ = D + CZ n ( I − AZ n ) − B, Z n = k M r =1 ( Z ( r ) ⊗ I n r ) . Therefore cg = D + C ( ρ − Z n )( I − A ( ρ − Z n )) − B = D + ρ − CZ n ( I − ρ − AZ n ) − B. Then we lift the rational function cg to a noncommutative rational expression using the samerealization formula, D + C ′ z n ( I − A ′ z n )) − B, now with z n = L kr =1 ( z ( r ) ⊗ I n r ) and the entries z ( r ) ij of matrices z ( r ) being noncommutingindeterminates, r = 1 , . . . , k , i = 1 , . . . , ℓ r , j = 1 , . . . , m r , and A ′ = ρ − A , C ′ = ρ − C (cf.(2.1)). Notice that the colligation matrix (cid:2) A ′ BC ′ D (cid:3) is contractive, with k A ′ k < k C ′ k < R = D min + C min z n min (cid:16) I − A min z n min (cid:17) − B min , whose colligation matrix (cid:2) A min B min C min D min (cid:3) is still contractive and such that k A min k < k C min k <
1. By Proposition 2.3, we also obtain a minimal noncommutative structuredsystem realization of R − , R − = D × min + C × min z n min (cid:16) I − A × min z n min (cid:17) − B × min , ith the colligation matrix (cid:20) A × min B × min C × min D × min (cid:21) = (cid:20) A min − B min D − C min B min D − − D − C min D − (cid:21) (cf., (2.5)–(2.6)). Applying Theorem 2.1 to R − and Corollary 2.2 to R − = p/c , we obtainthat the singularity set of the polynomial p/c (which is the empty set), agrees with n Z = ( Z (1) , . . . , Z ( k ) ) ∈ C ℓ r × m r × · · · × C ℓ rk × m rk : det( I − A × min Z n min ) = 0 o , which is possible only if det( I − A × min Z n min ) ≡ (cid:20) I − A min Z n min B min − C min Z n min D min (cid:21) = (cid:20) I − C min Z n min ( I − A min Z n min ) − I (cid:21) (cid:20) I − A min Z n min c/p (cid:21) (cid:20) I ( I − A min Z n min ) − B I (cid:21) = (cid:20) I B × min I (cid:21) (cid:20) I − A × min Z n min D min (cid:21) (cid:20) I C × min Z n min I (cid:21) , we obtain thatdet (cid:20) I − A min Z n min B min − C min Z n min D min (cid:21) = cp det( I − A min Z n min ) = D min det( I − A × min Z n min )= D min = cp (0) = c. Therefore, p = det( I − A min Z n min ). Since A min is a strict contraction, we obtain that (3.1) istrue with K = A min and n min in the place of n . (cid:3) Corollary 3.2.
Every strongly D d -stable polynomial p is an eventual Agler denominator,i.e., there exists n = ( n , . . . , n d ) ∈ Z d + such that the rational inner function (3.2) z n ¯ p (1 /z ) p ( z ) is in the Schur–Agler class. Here for z = ( z , . . . , z d ) we set /z = (1 /z , . . . , /z d ) , ¯ p ( z ) = p (¯ z , . . . , ¯ z d ) , and z n = z n · · · z n d d .Proof. By Theorem 3.1 applied to the polydisk case, p has a strictly contractive determinantalrepresentation (3.1). By [12, Theorem 5.2], p is an eventual Agler denominator. Moreover, n in (3.2) can be chosen the same as in a (not necessarily strictly) contractive determinantalrepresentation (3.1) for p . (cid:3) References [1] J. Agler. On the representation of certain holomorphic functions defined on a polydisc, In Topics inoperator theory: Ernst D. Hellinger Memorial Volume,
Oper. Theory Adv. Appl. , Vol. , pp. 47–66,Birkh¨auser, Basel, 1990.[2] C.-G. Ambrozie and D. A. Timotin. Von Neumann type inequality for certain domains in C n . Proc.Amer. Math. Soc.
131 (2003), no. 3, 859–869 (electronic).[3] D. Z. Arov. Passive linear steady-state dynamical systems. (Russian)
Sibirsk. Mat. Zh.
20 (1979), no.2, 211–228, 457.
4] J. A. Ball and V. Bolotnikov. Realization and interpolation for Schur–Agler-class functions on domainswith matrix polynomial defining function in C n . J. Funct. Anal.
213 (2004), no. 1, 45–87.[5] J. A. Ball, G. Groenewald, and T. Malakorn. Structured noncommutative multidimensional linear sys-tems.
SIAM J. Control Optim.
44 (2005), no. 4, 1474–1528.[6] J. A. Ball and D. S. Kaliuzhnyi-Verbovetskyi. Rational Cayley inner Herglotz–Agler functions: Positive-kernel decompositions and transfer-function realizations.
Linear Algebra Appl.
456 (2014), 138–156.[7] S. Basu and A. Fettweis. New results on stable multidimensional polynomials. II. Discrete case.
IEEETrans. Circuits and Systems
34 (1987), 1264–1274.[8] J. Borcea, P. Br¨and´en, and T. M. Liggett. Negative dependence and the geometry of polynomials.
J.Amer. Math. Soc.
22 (2009), no. 2, 521–567.[9] J. C. Doyle. Analysis of feedback systems with structured uncertainties.
Proc. IEE-D
129 (1982), no. 6,242–250.[10] A. Grinshpan, D. S. Kaliuzhnyi-Verbovetskyi, V. Vinnikov, and H. J. Woerdeman. Stable and real-zeropolynomials in two variables.
Multidim. Syst. Sign. Process.
Published online.[11] A. Grinshpan, D. S. Kaliuzhnyi-Verbovetskyi, V. Vinnikov, and H. J. Woerdeman. Matrix-valued Her-mitian Positivstellensatz, lurking contractions, and contractive determinantal representations of stablepolynomials. Preprint, arXiv 1501.05527.[12] A. Grinshpan, D. S. Kaliuzhnyi-Verbovetskyi, and H. J. Woerdeman. Norm-constrained determinantalrepresentations of multivariable polynomials.
Complex Anal. Oper. Theory
Electron. J. Combin. , 15 (2008), no. 1, ResearchPaper 66, 26 pp.[15] D. S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov, Foundations of Free Non-commutative Function Theory.Math Surveys and Monographs, Vol. 199, AMS, 2014, 183 pp.[16] D. S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov, Noncommutative rational functions, their difference-differential calculus and realizations.
Multidimens. Syst. Signal Process.
23 (2012), no. 1–2, 49–77.[17] D. S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov. Singularities of rational functions and minimal factor-izations: The noncommutative and the commutative setting.
Linear Algebra Appl.
430 (2009), no. 4,869–889.[18] G. Knese. Rational inner functions in the Schur-Agler class of the polydisk.
Publ. Mat. , 55 (2011),343–357.[19] A. Kummert. 2-D stable polynomials with parameter-dependent coefficients: generalizations and newresults.
IEEE Trans. Circuits Systems I: Fund. Theory Appl.
49 (2002), 725–731.[20] A. Kummert. Synthesis of two-dimmensional lossless m -ports with prescribed scattering matrix. CircuitsSystems Signal Processing k -variable Schur polynomials. IEEE Trans. Circuits Sys-tems
37 (1990), no. 10, 1288–1291.[22] L. Li , L. Xu, and Z. Lin. Stability and stabilisation of linear multidimensional discrete systems in thefrequency domain.
Int. J. Control
86 (2013), no. 11, 1969–1989.[23] M. Scheicher. Robustly stable multivariate polynomials.
Multidimens. Syst. Signal Process.
24 (2013),no. 1, 23–50.[24] J. L. Taylor. A joint spectrum for several commuting operators.
J. Functional Analysis
Acta Math.
125 (1970),1–38.[26] D. G. Wagner. Multivariate stable polynomials: theory and applications.
Bull. Amer. Math. Soc. epartment of Mathematics, Drexel University, 3141 Chestnut St., Philadelphia, PA,19104 E-mail address : { tolya,dmitryk,hugo } @math.drexel.edu Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel,84105
E-mail address : [email protected]@math.bgu.ac.il