Contractive determinantal representations of stable polynomials on a matrix polyball

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

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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 languageEnglish
Pages (from-to)25-37
Number of pages13
JournalMathematische Zeitschrift
Volume283
Issue number1-2
DOIs
StatePublished - 1 Jun 2016

Keywords

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

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