Transfer-function realization for multipliers of the Arveson space

Abstract

An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers for the reproducing kernel Hilbert space $\mathcal{H}(k_d)$ on the unit ball ${\mathbb{B}}^d\subset {\mathbb{C}}^d$, where $k_d$ is the positive kernel $k_d(\lambda ,\zeta )=1/(1-<\lambda ,\zeta >)$ on ${\mathbb{B}}^d$. We study this space from the point of view of realization theory and functional models of de Branges-Rovnyak type. We highlight features which depart from the classical univariate case: coisometric realizations have only partial uniqueness properties, the nonuniqueness can be described explicitly, and this description assumes a particularly concrete form in the functional-model context.

References in corpus (2)

• Multivariable backward-shift-invariant subspaces and observability operators
• Schur-class multipliers on the Arveson space: de Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations