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Contractive determinantal representations of stable polynomials on a matrix polyball

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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.

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Notes

  1. Here and in the rest of the paper we use a convention that a matrix block which involves \(I_{n_i}\) is void in the case of \(n_i\) equal to 0.

  2. We note that, even without a contractiveness requirement for K, constructing a representation with \(n=\deg p\) is, in general, impossible, because it involves solving an overdetermined system of equations for the entries of K when n is sufficiently large.

  3. Here, similarly to the convention we made in a footnote on the front page of the paper, we assume that a matrix block is void if the number of its rows/columns is 0.

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Authors and Affiliations

  1. Department of Mathematics, Drexel University, 3141 Chestnut St, Philadelphia, PA, 19104, USA

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

  2. Department of Mathematics, Ben-Gurion University of the Negev, 84105, Beer-Sheva, Israel

    Victor Vinnikov

Authors
  1. Anatolii Grinshpan
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  2. Dmitry S. Kaliuzhnyi-Verbovetskyi
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  3. Victor Vinnikov
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  4. Hugo J. Woerdeman
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Corresponding author

Correspondence to Dmitry S. Kaliuzhnyi-Verbovetskyi.

Additional information

AG, DK-V, HW were partially supported by NSF Grant DMS-0901628. DK-V and VV were partially supported by BSF Grant 2010432.

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Grinshpan, A., Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V. et al. Contractive determinantal representations of stable polynomials on a matrix polyball. Math. Z. 283, 25–37 (2016). https://doi.org/10.1007/s00209-015-1587-4

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  • DOI: https://doi.org/10.1007/s00209-015-1587-4

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