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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
MathSciNet review: 2929013
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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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L. A. Coburn
Affiliation: Department of Mathematics, The State University of New York at Buffalo, Buffalo, New York 14260

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.