Analytic models for commuting operator tuples on bounded symmetric domains

摘要

For a domain Omega in C-d and a Hilbert space H of analytic functions on Omega which satisfies certain conditions, we characterize the commuting d-tuples T = (T-1,..., T-d) of operators on a separable Hilbert space H such that T* is unitarily equivalent to the restriction of M* to an invariant subspace, where M is the operator d-tuple Z x I on the Hilbert space tensor product H x H. For Omega the unit disc and H the Hardy space H-2, this reduces to a well-known theorem of Sz.-Nagy and Foias; for H a reproducing kernel Hilbert space on Omega subset of C-d such that the reciprocal 1/K (x, (y) over bar) of its reproducing kernel is a polynomial in x and (y) over bar, this is a recent result of Ambrozie, Muller and the second author. In this paper, we extend the latter result by treating spaces H for which 1/K ceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation) H = H-nu on a Cartan domain corresponding to the parameter nu in the continuous Wallach set, and reproducing kernel Hilbert spaces H for which 1/K is a rational function. Further, we treat also the more general problem when the operator M is replaced by M + W, W being a certain generalization of a unitary operator tuple. For the case of the spaces H-nu on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on Omega, which seems to be of an independent interest. [References: 30]