The Berezin Transform and Radial Operators on the Bergman Space of the Unit Ball

In this paper, we investigate the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a class of operators that we call radial operators, an oscillation criterion and diagonal are sufficient conditions...

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Bibliographic Details
Published in:Complex analysis and operator theory 2013-02, Vol.7 (1), p.313-329
Main Authors: Zhou, Ze-Hua, Chen, Wei-Li, Dong, Xing-Tang
Format: Article
Language:English
Subjects:
Analysis
Mathematics
Mathematics and Statistics
Operator Theory
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Summary:In this paper, we investigate the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a class of operators that we call radial operators, an oscillation criterion and diagonal are sufficient conditions under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit sphere. We further study a special class of radial operators, i.e., Toeplitz operators with a radial L 1 ( B n ) symbol.
ISSN:1661-8254
1661-8262
DOI:10.1007/s11785-011-0145-2