Matrix-valued Hermitian Positivstellensatz, Lurking Contractions, and Contractive Determinantal Representations of Stable Polynomials

Grinshpan, Anatolii; Kaliuzhnyi-Verbovetskyi, Dmitry S.; Vinnikov, Victor; Woerdeman, Hugo J.

doi:10.1007/978-3-319-31383-2_7

Cited by 9 publications

(9 citation statements)

References 13 publications

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“…The existence of a representation (1.1) with a contractive (resp., strictly contractive) matrix 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., strong stability) of p is guaranteed. In a recent paper of the authors [11], the following result has been obtained. Let P ∈ C ℓ×m [z 1 , .…”

Section: Introductionmentioning

confidence: 82%

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Contractive determinantal representations of stable polynomials on a matrix polyball

Grinshpan

Kaliuzhnyi-Verbovetskyi

Vinnikov

et al. 2015

Math. Z.

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Abstract. We show that an irreducible polynomial p with no zeros on the closure of a matrix 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 1 , . . . , n k ) is a k-tuple of nonnegative integers,ij ] are complex matrices, p is a polynomial in the matrix entries z (r) ij , and K is a strictly contractive matrix. This result is obtained via a noncommutative lifting and a theorem on the singularities of minimal noncommutative structured system realizations.

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Section: Introductionmentioning

confidence: 82%

“…Thus we can find a constant c > 0 so that cg ρ A,Z < 1. By [11,Theorem 3.4] applied to F = cg ρ , we obtain a n = (n 1 , . .…”

Section: Contractive Determinantal Depresentations Of Stable Polynomimentioning

confidence: 99%

Contractive determinantal representations of stable polynomials on a matrix polyball

Grinshpan

Kaliuzhnyi-Verbovetskyi

Vinnikov

et al. 2015

Math. Z.

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“…Theorem 5.1 is somewhat unsatisfying since the kernel decomposition (5.3) holds only for Z, W ∈ Ω even if we assume that the kernel S is given on all of Ξ, unlike the case for Positivstellensätze as appearing in classical commutative algebraic geometry (see e.g. [13,21] as well as [15] for the matrix-valued case). To get a version of Theorem The result of Theorem 5.1 is that Problem C ′ has an affirmative solution under some additional hypotheses, but in a weaker form.…”

Section: Problem C: Kernel-dominance Certificatesmentioning

confidence: 99%

Free noncommutative hereditary kernels: Jordan decomposition, Arveson extension, kernel domination

Ball¹,

Marx²,

Vinnikov³

2022

Preprint

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We discuss a (i) quantized version of the Jordan decomposition theorem for a complex Borel measure on a compact Hausdorff space, namely, the more general problem of decomposing a general noncommutative kernel (a quantization of the standard notion of kernel function) as a linear combination of completely positive noncommutative kernels (a quantization of the standard notion of positive definite kernel). Other special cases of (i) include: the problem of decomposing a general operator-valued kernel function as a linear combination of positive kernels (not always possible), of decomposing a general bounded linear Hilbert-space operator as a linear combination of positive linear operators (always possible), of decomposing a completely bounded linear map from a C * -algebra A to an injective C * -algebra L(Y) as a linear combination of completely positive maps from A to L(Y) (always possible). We also discuss (ii) a noncommutative kernel generalization of the Arveson extension theorem (any completely positive map φ from a operator system S to an injective C * -algebra L(Y) can be extended to a completely positive map φe from a C * -algebra containing S to L(Y)), and (iii) a noncommutative kernel version of a Positivstellensatz (i.e., finding a certificate to explain why one kernel is positive at points where another given kernel is positive).

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“…. , z d ) ∈ C d satisfying P(z) < 1 (see [2,4,11]); in this case, P = Z viewed as a polynomial in matrix entries z (r) ij , i, j = 1, . .…”

Section: Introductionmentioning

confidence: 99%

Rational Inner Functions on a Square-Matrix Polyball

Grinshpan

Kaliuzhnyi-Verbovetskyi

Vinnikov

et al. 2017

Association for Women in Mathematics Series

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Dedicated to the blessed memory of Cora Sadosky, our dear friend and colleague.Abstract. We establish the existence of a finite-dimensional unitary realization for every matrix-valued rational inner function from the Schur-Agler class on a unit square-matrix polyball. In the scalar-valued case, we characterize the denominators of these functions. We also show that every polynomial with no zeros in the closed domain is such a denominator. One of our tools is the Korányi-Vagi theorem generalizing Rudin's description of rational inner functions to the case of bounded symmetric domains; we provide a short elementary proof of this theorem suitable in our setting.

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Matrix-valued Hermitian Positivstellensatz, Lurking Contractions, and Contractive Determinantal Representations of Stable Polynomials

Cited by 9 publications

References 13 publications

Contractive determinantal representations of stable polynomials on a matrix polyball

Contractive determinantal representations of stable polynomials on a matrix polyball

Free noncommutative hereditary kernels: Jordan decomposition, Arveson extension, kernel domination

Rational Inner Functions on a Square-Matrix Polyball

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