Contractive determinantal representations of stable polynomials on a matrix polyball

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

doi:10.1007/s00209-015-1587-4

Contractive determinantal representations of stable polynomials on a matrix polyball

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

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

We show that a 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 (Formula presented.) , admits a strictly contractive determinantal representation, i.e., (Formula presented.) , where (Formula presented.) is a k-tuple of nonnegative integers, (Formula presented.) , (Formula presented.) are complex matrices, p is a polynomial in the matrix entries (Formula presented.) , 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.

Original language	English
Pages (from-to)	25-37
Number of pages	13
Journal	Mathematische Zeitschrift
Volume	283
Issue number	1-2
DOIs	https://doi.org/10.1007/s00209-015-1587-4
State	Published - 1 Jun 2016

Keywords

Classical Cartan domain
Contractive determinantal representation
Contractive realization
Polyball
Stable polynomial
Structured noncommutative multidimensional system

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/s00209-015-1587-4

Cite this

@article{b4ecc2d175b34ce08666544cd7552888,

title = "Contractive determinantal representations of stable polynomials on a matrix polyball",

abstract = "We show that a 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 (Formula presented.) , admits a strictly contractive determinantal representation, i.e., (Formula presented.) , where (Formula presented.) is a k-tuple of nonnegative integers, (Formula presented.) , (Formula presented.) are complex matrices, p is a polynomial in the matrix entries (Formula presented.) , 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.",

keywords = "Classical Cartan domain, Contractive determinantal representation, Contractive realization, Polyball, Stable polynomial, Structured noncommutative multidimensional system",

author = "Anatolii Grinshpan and Kaliuzhnyi-Verbovetskyi, {Dmitry S.} and Victor Vinnikov and Woerdeman, {Hugo J.}",

note = "Publisher Copyright: {\textcopyright} 2015, Springer-Verlag Berlin Heidelberg.",

year = "2016",

month = jun,

day = "1",

doi = "10.1007/s00209-015-1587-4",

language = "English",

volume = "283",

pages = "25--37",

journal = "Mathematische Zeitschrift",

issn = "0025-5874",

publisher = "Springer New York",

number = "1-2",

}

TY - JOUR

T1 - Contractive determinantal representations of stable polynomials on a matrix polyball

AU - Grinshpan, Anatolii

AU - Kaliuzhnyi-Verbovetskyi, Dmitry S.

AU - Vinnikov, Victor

AU - Woerdeman, Hugo J.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - We show that a 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 (Formula presented.) , admits a strictly contractive determinantal representation, i.e., (Formula presented.) , where (Formula presented.) is a k-tuple of nonnegative integers, (Formula presented.) , (Formula presented.) are complex matrices, p is a polynomial in the matrix entries (Formula presented.) , 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.

AB - We show that a 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 (Formula presented.) , admits a strictly contractive determinantal representation, i.e., (Formula presented.) , where (Formula presented.) is a k-tuple of nonnegative integers, (Formula presented.) , (Formula presented.) are complex matrices, p is a polynomial in the matrix entries (Formula presented.) , 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.

KW - Classical Cartan domain

KW - Contractive determinantal representation

KW - Contractive realization

KW - Polyball

KW - Stable polynomial

KW - Structured noncommutative multidimensional system

UR - http://www.scopus.com/inward/record.url?scp=84947791229&partnerID=8YFLogxK

U2 - 10.1007/s00209-015-1587-4

DO - 10.1007/s00209-015-1587-4

M3 - Article

AN - SCOPUS:84947791229

SN - 0025-5874

VL - 283

SP - 25

EP - 37

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 1-2

ER -

Contractive determinantal representations of stable polynomials on a matrix polyball

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this