Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Toeplitz operators with distributional symbols on Bergman spaces

Antti Peräläa1, Jari Taskinena1 and Jani Virtanena2

a1 Department of Mathematics, University of Helsinki, 00014 Helsinki, Finland (antti.i.perala@helsinki.fi; jari.taskinen@helsinki.fi)

a2 Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA (virtanen@courant.nyu.edu)

Abstract

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces $A^p(\mathbb{D})$, 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space $\smash{W_\nu^{-m,\infty}(\mathbb{D})}$ 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.

(Received February 01 2010)

(Online publication April 07 2011)

Keywords

  • Toeplitz operator;
  • Bergman space;
  • distribution;
  • bounded operator;
  • compact operator

2010 Mathematics subject classification

  • Primary 47B35