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 $K$.
Categories: math.CV math.CA math.RA