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Toeplitz operators with distributional symbols on Bergman spaces

Part of: Special classes of linear operators

Published online by Cambridge University Press:  07 April 2011

Antti Perälä ,
Jari Taskinen  and
Jani Virtanen
Antti Perälä
Affiliation:
Department of Mathematics, University of Helsinki, 00014 Helsinki, Finland (antti.i.perala@helsinki.fi; jari.taskinen@helsinki.fi)
Jari Taskinen
Affiliation:
Department of Mathematics, University of Helsinki, 00014 Helsinki, Finland (antti.i.perala@helsinki.fi; jari.taskinen@helsinki.fi)
Jani Virtanen
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA (virtanen@courant.nyu.edu)
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Abstract

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We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

Keywords

Toeplitz operatorBergman spacedistributionbounded operatorcompact operator

MSC classification

Secondary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 2 , June 2011 , pp. 505 - 514
Copyright
Copyright © Edinburgh Mathematical Society 2011

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