Matrix-valued Hermitian Positivstellensatz, Lurking Contractions, and Contractive Determinantal Representations of Stable Polynomials

Abstract

We prove that every matrix-valued rational function F, which is regular on the closure of a bounded domain $\mathcal{D}_{p}\; \mathrm{in}\;\mathbb{C}^{d}$ and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realization

$$F(z)\;=\;D\;+\;CP(z)_{n}(I-AP(z)_n)^{-1}B$$

.

Here $\mathcal{D}_{p}$ is defined by the inequality $\|\mathrm{P}(z)\|\;<\;1$, where $\mathrm{P}(z)$ is a direct sum of matrix polynomials $\mathrm{P_i}(z)$ (so that an appropriate Archimedean condition is satisfied), and $\mathrm{P}(z)_n\;=\;\oplus^k_{i=1}\mathrm{P_i}(z)\otimes\;I_{n_{i}}$, 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 $\mathcal{D}_{p}$ is a factor of det $(1-KP(z)_{n})$, with a contractive matrix K.