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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Berezin transform on real bounded symmetric domains


Author: Genkai Zhang
Journal: Trans. Amer. Math. Soc. 353 (2001), 3769-3787
MSC (2000): Primary 22E46, 43A85, 32M15, 53C35
Published electronically: May 4, 2001
MathSciNet review: 1837258
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Abstract:

Let $\mathbb D$ be a bounded symmetric domain in a complex vector space $V_{\mathbb C}$with a real form $V$ and $D=\mathbb D\cap V=G/K$ be the real bounded symmetric domain in the real vector space $V$. We construct the Berezin kernel and consider the Berezin transform on the $L^2$-space on $D$. The corresponding representation of $G$is then unitarily equivalent to the restriction to $G$of a scalar holomorphic discrete series of holomorphic functions on $\mathbb D$ and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the $L^2$-space.


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Additional Information

Genkai Zhang
Affiliation: Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden
Email: genkai@math.chalmers.se

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02832-X
PII: S 0002-9947(01)02832-X
Keywords: Real bounded symmetric domains, Jordan triples, Siegel domains, Berezin transform, invariant differential operators, unitary representations of Lie groups, irreducible decomposition
Received by editor(s): January 16, 2000
Received by editor(s) in revised form: October 10, 2000
Published electronically: May 4, 2001
Additional Notes: Research supported by the Swedish Natural Sciences Research Council (NFR)
Article copyright: © Copyright 2001 American Mathematical Society