Analytic models for commuting operator tuples on bounded symmetric domains

Jonathan Arazy, Miroslav Engliš

Research output: Contribution to journalArticlepeer-review

Abstract

For a domain Ω in ℂd 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. 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 ⊂ ℂd 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 = ℋv 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 ℋv 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 languageEnglish
Pages (from-to)837-864
Number of pages28
JournalTransactions of the American Mathematical Society
Volume355
Issue number2
StatePublished - Feb 2003

Keywords

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

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics

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