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On Toeplitz operators on Bergman spaces of the unit polydisk

Author: Trieu Le
Journal: Proc. Amer. Math. Soc. 138 (2010), 275-285
MSC (2000): Primary 47B35
Published electronically: August 25, 2009
MathSciNet review: 2550193
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Abstract: We study Toeplitz operators on the Bergman space $ A^2_{\vartheta}$ of the unit polydisk $ \mathbb{D}^n$, where $ \vartheta$ is a product of $ n$ rotation-invariant regular Borel probability measures. We show that if $ f$ is a bounded Borel function on $ \mathbb{D}^n$ such that $ F(w)=\lim\limits_{\substack{z\rightarrow w\\ z\in\mathbb{D}^n}}f(z)$ exists for all $ w\in\partial\mathbb{D}^n$, then $ T_f$ is compact if and only if $ F=0$ a.e. with respect to a measure $ \gamma$ associated with $ \vartheta$ on the boundary $ \partial\mathbb{D}^n$ . We also discuss the commuting problem: if $ g$ is a non-constant bounded holomorphic function on $ \mathbb{D}^n$, then what conditions does a bounded function $ f$ need to satisfy so that $ T_f$ commutes with $ T_g$?

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Trieu Le
Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1

Keywords: Bergman space, Toeplitz operator, compact operator, commuting problem
Received by editor(s): November 9, 2008
Received by editor(s) in revised form: May 28, 2009
Published electronically: August 25, 2009
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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