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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Berezin transform and Weyl-type unitary operators on the Bergman space


Author: L. A. Coburn
Journal: Proc. Amer. Math. Soc. 140 (2012), 3445-3451
MSC (2010): Primary 47B32; Secondary 32A36
Published electronically: February 15, 2012
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Abstract: For $ \mathbf {D}$ the open complex unit disc with normalized area measure, we consider the Bergman space $ L_{a}^{2}(\mathbf {D})$ of square-integrable holomorphic functions on $ \mathbf {D}$. Induced by the group $ Aut(\mathbf {D})$ of biholomorphic automorphisms of $ \mathbf {D}$, there is a standard family of Weyl-type unitary operators on $ L_{a}^{2}(\mathbf {D})$. For all bounded operators $ X$ on $ L_{a}^{2}(\mathbf {D})$, the Berezin transform $ \widetilde X$ is a smooth, bounded function on $ \mathbf {D}$. The range of the mapping Ber: $ X \rightarrow \widetilde X$ is invariant under $ Aut(\mathbf {D} )$. The ``mixing properties'' of the elements of $ Aut(\mathbf {D} )$ are visible in the Berezin transforms of the induced unitary operators. Computations involving these operators show that there is no real number $ M>0$ with $ M\Vert \widetilde X \Vert _{\infty } \geq \Vert X \Vert $ for all bounded operators $ X$ and are used to check other possible properties of $ \widetilde X$. Extensions to other domains are discussed.


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Additional Information

L. A. Coburn
Affiliation: Department of Mathematics, The State University of New York at Buffalo, Buffalo, New York 14260
Email: lcoburn@buffalo.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11440-6
PII: S 0002-9939(2012)11440-6
Received by editor(s): April 5, 2011
Published electronically: February 15, 2012
Communicated by: Richard Rochberg
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.