The Berezin transform and radial operators

Zorboska, Nina

doi:10.1090/S0002-9939-02-06691-1

The Berezin transform and radial operators
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by Nina Zorboska

Proc. Amer. Math. Soc. 131 (2003), 793-800

DOI: https://doi.org/10.1090/S0002-9939-02-06691-1

Published electronically: July 2, 2002

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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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