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

  • Anatolii GrinshpanEmail author
  • Dmitry S. Kaliuzhnyi-Verbovetskyi
  • Victor Vinnikov
  • Hugo J. Woerdeman
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 255)

Abstract

We prove that every matrix-valued rational function F, which is regular on the closure of a bounded domain \(\mathcal{D}_{p}\; \mathrm{in}\;\mathbb{C}^{d}\) and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realization
$$F(z)\;=\;D\;+\;CP(z)_{n}(I-AP(z)_n)^{-1}B$$
.

Here \(\mathcal{D}_{p}\) is defined by the inequality \(\|\mathrm{P}(z)\|\;<\;1\), where \(\mathrm{P}(z)\) is a direct sum of matrix polynomials \(\mathrm{P_i}(z)\) (so that an appropriate Archimedean condition is satisfied), and \(\mathrm{P}(z)_n\;=\;\oplus^k_{i=1}\mathrm{P_i}(z)\otimes\;I_{n_{i}}\), with some k-tuple n of multiplicities n i; 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 \(\mathcal{D}_{p}\) is a factor of det \((1-KP(z)_{n})\), with a contractive matrix K.

Keywords

Polynomially defined domain classical Cartan domains contractive realization determinantal representation multivariable polynomial stable polynomial 

Mathematics Subject Classification (2010).

15A15 47A13 13P15 90C25 93B28 47N70 

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Anatolii Grinshpan
    • 1
    Email author
  • Dmitry S. Kaliuzhnyi-Verbovetskyi
    • 1
  • Victor Vinnikov
    • 2
  • Hugo J. Woerdeman
    • 1
  1. 1.Department of MathematicsDrexel UniversityPhiladelphiaUSA
  2. 2.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael

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