Matrix-valued hermitian positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials

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

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

We prove that every matrix-valued rational function F, which is regular on the closure of a bounded domain DP in Cd and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realization [Formula presented]. Here DP is defined by the inequality [Formula presented] where P (z) is a direct sum of matrix polynomials Pi(z) (so that an appropriate Archimedean condition is satisfied), and [Formula presented], with some k-tuple n of multiplicities ni; special cases include the open unit polydisk and the classical Cartan domains. The proof uses a matrix-valued version of a Hermitian Positivstellensatz by Putinar, and a lurking contraction argument. As a consequence, we show that every polynomial with no zeros on the closure of DP is a factor of det(I − KP(z)n), with a contractive matrix K.

Original languageEnglish
Pages (from-to)123-136
Number of pages14
JournalOperator Theory: Advances and Applications
Volume255
DOIs
StatePublished - 1 Jan 2016

Keywords

  • Classical Cartan domains
  • Contractive realization
  • Determinantal representation
  • Multivariable polynomial
  • Polynomially defined domain
  • Stable polynomial

ASJC Scopus subject areas

  • Analysis

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