Contractive determinantal representations of stable polynomials on a matrix polyball

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., $p=\det (I-K{Z}_n)$, where $n=(n_1,...,n_k)$ is a $k$-tuple of nonnegative integers, $Z_n={⨁}_{r=1}^k(Z^{(r)}\otimes I_{n_r})$, $Z^{(r)}=[z_{ij}^{(r)}]$ are complex matrices, $p$ is a polynomial in the matrix entries $z_{ij}^{(r)}$, 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.