Abstract. 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,...,Td) 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. ForΩ 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 Ω ⊂ C d such that the reciprocal 1/K(x, y) of its reproducing kernel is a polynomial in x and y, 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 = Hν on a Cartan domain corresponding to the parameter ν 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
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