## 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,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.

Original language | English |
---|---|

Pages (from-to) | 837-864 |

Number of pages | 28 |

Journal | Transactions of the American Mathematical Society |

Volume | 355 |

Issue number | 2 |

State | Published - Feb 2003 |

## Keywords

- Bounded symmetric domains
- Coanalytic models
- Commuting operator tuples
- Functional calculus
- Reproducing kernels

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics