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.

N1 - Funding Information:
AG, DK-V, HW were partially supported by NSF Grant DMS-0901628. DK-V and VV were partially supported by BSF Grant 2010432.
Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

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

VL - 283

SP - 25

EP - 37

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 1-2

ER -