Berezin transform and Weyl-type unitary operators on the Bergman space
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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.References
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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
- Received by editor(s): April 5, 2011
- Published electronically: February 15, 2012
- Communicated by: Richard Rochberg
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3445-3451
- MSC (2010): Primary 47B32; Secondary 32A36
- DOI: https://doi.org/10.1090/S0002-9939-2012-11440-6
- MathSciNet review: 2929013