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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Published electronically: October 9, 2002
MathSciNet review: 1932728
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Abstract: For a domain $\Omega $ in $\mathbb{C} ^{d}$ and a Hilbert space $\mathcal{H} $of analytic functions on $\Omega $ which satisfies certain conditions, we characterize the commuting $d$-tuples $T=(T_{1},\dots ,T_{d})$ 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 $Z\otimes I$ on the Hilbert space tensor product  $\mathcal{H} \otimes H$. For $\Omega $ the unit disc and $\mathcal{H} $ the Hardy space $H^{2}$, this reduces to a well-known theorem of Sz.-Nagy and Foias; for $\mathcal{H} $a reproducing kernel Hilbert space on $\Omega \subset \mathbb{C} ^{d}$ such that the reciprocal $1/K(x,\overline{y}) $ of its reproducing kernel is a polynomial in $x$ and  $\overline y $, this is a recent result of Ambrozie, Müller and the second author. In this paper, we extend the latter result by treating spaces $\mathcal{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) $\mathcal{H} =\mathcal{H} _{\nu }$ on a Cartan domain corresponding to the parameter $\nu $ in the continuous Wallach set, and reproducing kernel Hilbert spaces $\mathcal{H} $ for which $1/K$ is a rational function. Further, we treat also the more general problem when the operator $M$is replaced by $M\oplus W$, $W$ being a certain generalization of a unitary operator tuple. For the case of the spaces $\mathcal{H} _{\nu }$ on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on $\Omega $, 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

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

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