Schur-class multipliers on the Arveson space: de Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations

Abstract

An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers $S$ 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 ${\mathbb{B}}^d$. The reproducing kernel space $\mathcal{H}(K_S)$ associated with the positive kernel $K_S(\lambda ,\zeta )=(I-S(\lambda )S(\zeta {)}^{\ast })\cdot k_d(\lambda ,\zeta )$ is a natural multivariable generalization of the classical de Branges-Rovnyak canonical model space. A special feature appearing in the multivariable case is that the space $\mathcal{H}(K_S)$ in general may not be invariant under the adjoints $M_{{\lambda }_j}^{\ast }$ of the multiplication operators $M_{{\lambda }_j}:f(\lambda )\mapsto {\lambda }_{j}f(\lambda )$ on $\mathcal{H}(k_d)$. We show that invariance of $\mathcal{H}(K_S)$ under $M_{{\lambda }_j}^{\ast }$ for each $j=1,...,d$ is equivalent to the existence of a weakly coisometric realization for $S$ of the form $S(\lambda )=D+C(I-{\lambda }_1{A}_1...-{\lambda }_d{A}_d{)}^{-1}({\lambda }_1{B}_1+...+{\lambda }_d{B}_d)$ such that the state operators $A_1,...,A_d$ pairwise commute. We show that this special situation always occurs for the case of inner functions $S$ (where the associated multiplication operator $M_S$ is a partial isometry), and that inner multipliers are characterized by the existence of such a realization such that the state operators $A_1,>...,A_d$ satisfy an additional stability property.

References in corpus (1)

• Multivariable backward-shift-invariant subspaces and observability operators

Cited by in corpus (1)

• Transfer-function realization for multipliers of the Arveson space