Finite-dimensional contractive realizations on polynomially defined domains and determinantal representations of multivariable polynomials Academic Article uri icon

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 an appropriate Archimedean condition is 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 of types I--III. 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 $K$. When $\mathcal{D}_\mathbf{P}$ is the open unit polydisk, we show that a polynomial with no zeros in the domain is the denominator of a rational inner function of the Schur--Agler class if and only if it admits a contractive determinantal representation up to an almost self-reversive factor. We also show that every rational inner function which is regular on the closed unit polydisk can be multiplied with another such function so that the product is in the Schur--Agler class.

publication date

  • January 1, 2015