Analytic models for commuting operator tuples on bounded symmetric domains

For a domain Ω in ℂ^d and a Hilbert space ℋ of analytic functions on Ω which satisfies certain conditions, we characterize the commuting d-tuples T = (T₁,...,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 ⊗ I on the Hilbert space tensor product ℋ ⊗ H. For Ω the unit disc and ℋ the Hardy space H², this reduces to a well-known theorem of Sz.-Nagy and Foias; for ℋ a reproducing kernel Hilbert space on Ω ⊂ ℂ^d such that the reciprocal 1/K (x,ȳ) of its reproducing kernel is a polynomial in x and ȳ, this is a recent result of Ambrozie, Müller 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_v on a Cartan domain corresponding to the parameter v 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_v on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on Ω, which seems to be of an independent interest.