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

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 $K_{S}(\lambda, \zeta) = (I - S(\lambda) S(\zeta)^{*}) \cdot k_{d}(\lambda, \zeta)$ 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 $M_{\lambda_{j}}^{*}$ of the multiplication operators $M_{\lambda_{j}} \colon f(\lambda) \mapsto \lambda_{j} f(\lambda)$ on ${\mathcal H}(k_{d})$. We show that invariance of ${\mathcal H}(K_{S})$ under $M_{\lambda_{j}}^{*}$ 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.