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Schur-class multipliers on the Arveson space: de Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations | | Joseph A. Ball
; Vladimir Bolotnikov
; Quanlei Fang
; | | Date: |
21 Oct 2006 | | Subject: | Classical Analysis and ODEs | | 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 $K_{S}(lambda, zeta) = (I - S(lambda) S(zeta)^{*}) cdot k_{d}(lambda, zeta)$ 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. | | Source: | arXiv, math/0610638 | | Services: | Forum | Review | PDF | Favorites | |
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