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The Berezin transform and radial operators

Author: Nina Zorboska
Journal: Proc. Amer. Math. Soc. 131 (2003), 793-800
MSC (2000): Primary 47B37, 47B10; Secondary 47B35
Published electronically: July 2, 2002
MathSciNet review: 1937440
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Abstract: We analyze the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a special class of operators that we call radial operators, an oscilation criterion is a sufficient condition under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit circle. We further study a special class of radial operators, i.e., Toeplitz operators with a radial $L^{1}(\mathbb{D})$ symbol.

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Additional Information

Nina Zorboska
Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

Keywords: Berezin transform, radial operator, Toeplitz operator
Received by editor(s): February 6, 2001
Received by editor(s) in revised form: October 12, 2001
Published electronically: July 2, 2002
Additional Notes: This work was supported by an NSERC grant
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society