Contractive determinantal representations of stable polynomials on a matrix polyball

TL;DR

The paper proves that certain stable polynomials on matrix polyballs have strictly contractive determinantal representations, extending understanding of polynomial representations in multivariable operator theory.

Contribution

It introduces a new class of polynomials with strictly contractive determinantal representations on matrix polyballs, using noncommutative lifting techniques.

Findings

Polynomials with no zeros on the closure of matrix polyballs admit such representations.

The representation involves a strictly contractive matrix K.

The approach uses noncommutative lifting and structured system realizations.

Abstract

We show that an irreducible 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 $n=(n_1,...,n_k)$ 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.