Matrix-valued hermitian positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials

We prove that every matrix-valued rational function F, which is regular on the closure of a bounded domain DP in Cd and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realization [Formula presented]. Here DP is defined by the inequality [Formula presented] where P (z) is a direct sum of matrix polynomials Pi(z) (so that an appropriate Archimedean condition is satisfied), and [Formula presented], with some k-tuple n of multiplicities ni; 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 D_P is a factor of det(I − KP(z)_n), with a contractive matrix K.