Matrix-valued Hermitian Positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials

TL;DR

This paper establishes a matrix-valued Positivstellensatz and contractive realizations for stable matrix-valued rational functions on certain domains, leading to new determinantal representations of zero-free polynomials.

Contribution

It introduces a matrix-valued Hermitian Positivstellensatz and a lurking contraction method to obtain finite-dimensional contractive realizations for stable rational functions.

Findings

Every stable matrix-valued rational function admits a finite-dimensional contractive realization.

Polynomials with no zeros on the domain boundary are factors of determinants involving contractive matrices.

The results generalize classical domains like the polydisk and Cartan domains.

Abstract

We prove that every matrix-valued rational function $F$, which is regular on the closure of a bounded domain $\mathcal{D}_\mathbf{P}$ in $\mathbb{C}^d$ and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realization $$F(z)= D + C\mathbf{P}(z)_n(I-A\mathbf{P}(z)_n)^{-1} B. $$ Here $\mathcal{D}_\mathbf{P}$ is defined by the inequality $\|\mathbf{P}(z)\|<1$, where $\mathbf{P}(z)$ is a direct sum of matrix polynomials $\mathbf{P}_i(z)$ (so that appropriate Archimedean and approximation conditions are satisfied), and $\mathbf{P}(z)_n=\bigoplus_{i=1}^k\mathbf{P}_i(z)\otimes I_{n_i}$, 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 $\mathcal{D}_\mathbf{P}$ is a factor of $\det (I - K\mathbf{P}(z)_n)$, with a contractive matrix $K$.