Analytic models for commuting operator tuples on bounded symmetric domains
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- by Jonathan Arazy and Miroslav Engliš PDF
- Trans. Amer. Math. Soc. 355 (2003), 837-864 Request permission
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.References
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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 Engliš
- Affiliation: MÚ AV ČR, Žitná 25, 11567 Prague 1, Czech Republic
- Email: englis@math.cas.cz
- 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.
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1932728