Analytic models for commuting operator tuples on bounded symmetric domains

Authors:
Jonathan Arazy and Miroslav Englis

Journal:
Trans. Amer. Math. Soc. **355** (2003), 837-864

MSC (2000):
Primary 47A45; Secondary 47A13, 32M15, 32A07

DOI:
https://doi.org/10.1090/S0002-9947-02-03156-2

Published electronically:
October 9, 2002

MathSciNet review:
1932728

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Abstract | References | Similar Articles | Additional Information

Abstract: For a domain in and a Hilbert space of analytic functions on which satisfies certain conditions, we characterize the commuting -tuples of operators on a separable Hilbert space such that is unitarily equivalent to the restriction of to an invariant subspace, where is the operator -tuple on the Hilbert space tensor product . For the unit disc and the Hardy space , this reduces to a well-known theorem of Sz.-Nagy and Foias; for a reproducing kernel Hilbert space on such that the reciprocal of its reproducing kernel is a polynomial in and , this is a recent result of Ambrozie, Müller and the second author. In this paper, we extend the latter result by treating spaces for which ceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation) on a Cartan domain corresponding to the parameter in the continuous Wallach set, and reproducing kernel Hilbert spaces for which is a rational function. Further, we treat also the more general problem when the operator is replaced by , being a certain generalization of a unitary operator tuple. For the case of the spaces on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on , which seems to be of an independent interest.

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Additional Information

**Jonathan Arazy**

Affiliation:
Department of Mathematics, University of Haifa, Haifa 31905, Israel

Email:
jarazy@math.haifa.ac.il

**Miroslav Englis**

Affiliation:
MÚ AV ČR, Žitná 25, 11567 Prague 1, Czech Republic

Email:
englis@math.cas.cz

DOI:
https://doi.org/10.1090/S0002-9947-02-03156-2

Keywords:
Coanalytic models,
reproducing kernels,
bounded symmetric domains,
commuting operator tuples,
functional calculus

Received by editor(s):
February 14, 2002

Published electronically:
October 9, 2002

Additional Notes:
The second author’s research was supported by GA ČR grant no. 201/00/0208 and GA AV ČR 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.

Article copyright:
© Copyright 2002
American Mathematical Society