Schur-class multipliers on the Arveson space: de Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations
Loading...
Date
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
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 $k_{d}(λ, ζ) = 1/(1 - < λ, ζ>)$ on ${\mathbb B}^{d}$. The reproducing kernel space ${\mathcal H}(K_{S})$ associated with the positive kernel $K_{S}(λ, ζ) = (I - S(λ) S(ζ)^{*}) \cdot k_{d}(λ, ζ)$ 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 $M_{λ_{j}}^{*}$ of the multiplication operators $M_{λ_{j}} \colon f(λ) \mapsto λ_{j} f(λ)$ on ${\mathcal H}(k_{d})$.
We show that invariance of ${\mathcal H}(K_{S})$ under $M_{λ_{j}}^{*}$ for each $j = 1, ..., d$ is equivalent to the existence of a weakly coisometric realization for $S$ of the form $S(λ) = D + C (I - λ_{1}A_{1} ... - λ_{d} A_{d})^{-1}(λ_{1}B_{1} + ... + λ_{d} B_{d})$ 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.