Mathematische Zeitschrift

, Volume 283, Issue 1–2, pp 25–37 | Cite as

Contractive determinantal representations of stable polynomials on a matrix polyball

  • Anatolii Grinshpan
  • Dmitry S. Kaliuzhnyi-Verbovetskyi
  • Victor Vinnikov
  • Hugo J. Woerdeman
Article
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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 \(p(0)=1\), admits a strictly contractive determinantal representation, i.e., \(p=\det (I-KZ_n)\), where \(n=(n_1,\ldots ,n_k)\) is a k-tuple of nonnegative integers, \(Z_n=\bigoplus _{r=1}^k(Z^{(r)}\otimes I_{n_r})\), \(Z^{(r)}=[z^{(r)}_{ij}]\) are complex matrices, p is a polynomial in the matrix entries \(z^{(r)}_{ij}\), 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

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

Mathematics Subject Classification

15A15 32A10 47N70 14A22 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Anatolii Grinshpan
    • 1
  • 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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