Transfer-function realization for multipliers of the Arveson space

Details Transfer-function realization for multipliers of the Arveson space Transfer-function realization for multipliers of the Arveson space

2006-10-20

arxiv.org/abs/math/0610637v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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.

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