TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Number 2, Pages 837–864 S 0002-9947(02)03156-2 Article electronically published on October 9, 2002 ANALYTIC MODELS FOR COMMUTING OPERATOR TUPLES ON BOUNDED SYMMETRIC DOMAINS JONATHAN ARAZY AND MIROSLAV ENGLI ˇ S 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 =(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 ⊗ I on the Hilbert space tensor product H⊗ H. For Ω 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, M¨ uller 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 Car- tan domain corresponding to the parameter ν in the continuous Wallach set, and reproducing kernel Hilbert spaces H for which 1/K is a rational func- tion. 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 Hν 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. Let T be a bounded linear operator on a separable Hilbert space H . A well-known result of Sz.-Nagy and Foias says that the following two assertions are equivalent: (a) I − TT ∗ ≥ 0 (i.e., T is a contraction) and T ∗n → 0 in the strong operator topology; (b) T ∗ is unitarily equivalent to the restriction of a backward shift of infinite multiplicity to an invariant subspace. (0.1) There is also a somewhat stronger form of this result which deals with the case when the second half of the condition (a) is not satisfied: namely, (0.2) T (equivalently, T ∗ ) is a contraction if and only if T ∗ is unitarily equivalent to the restriction to an invariant subspace of the direct sum of a backward shift of infinite multiplicity and a unitary operator. Received by the editors February 14, 2002. 2000 Mathematics Subject Classification. Primary 47A45; Secondary 47A13, 32M15, 32A07. Key words and phrases. Coanalytic models, reproducing kernels, bounded symmetric domains, commuting operator tuples, functional calculus. The second author’s research was supported by GA ˇ CR grant no. 201/00/0208 and GA AV ˇ CR grant no. A1019005. The second author was also partially supported by the Israeli Academy of Sciences during his visit to Haifa in January 2001, during which part of this work was done. c 2002 American Mathematical Society 837 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use