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On the derivatives of the Berezin transform

Authors: Miroslav Englis and Genkai Zhang
Journal: Proc. Amer. Math. Soc. 134 (2006), 2285-2294
MSC (2000): Primary 47B32; Secondary 32A36, 32M15
Published electronically: February 2, 2006
MathSciNet review: 2213701
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Abstract: Improving upon a recent result of L. Coburn and J. Xia, we show that for any bounded linear operator $ T$ on the Segal-Bargmann space, the Berezin transform of $ T$ is a function whose partial derivatives of all orders are bounded. Similarly, if $ T$ is a bounded operator on any one of the usual weighted Bergman spaces on a bounded symmetric domain, then the appropriately defined ``invariant derivatives'' of any order of the Berezin transform of $ T$ are bounded. Further generalizations are also discussed.

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Additional Information

Miroslav Englis
Affiliation: Mathematics Institute, Academy of Sciences of the Czech Republic, Žitná 25, 11567 Praha 1, Czech Republic

Genkai Zhang
Affiliation: Chalmers Tekniska Högskola/Göteborgs Universitet, 412 96 Göteborg, Sweden

Keywords: Bergman kernel, Berezin transform, bounded symmetric domain, invariant differential operator
Received by editor(s): December 23, 2004
Received by editor(s) in revised form: March 1, 2005
Published electronically: February 2, 2006
Additional Notes: The research of the first author was supported by GA AV ČR grant no. A1019304
The research of the second author was supported by the Swedish Science Council (VR)
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society

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