Berezin transform and Toeplitz operators on harmonic Bergman spaces

Boo Rim Choe, Kyesook Nam

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

For an operator which is a finite sum of products of finitely many Toeplitz operators on the harmonic Bergman space over the half-space, we study the problem: Does the boundary vanishing property of the Berezin transform imply compactness? This is motivated by the Axler-Zheng theorem for analytic Bergman spaces, but the answer would not be yes in general, because the Berezin transform annihilates the commutator of any pair of Toeplitz operators. Nevertheless we show that the answer is yes for certain subclasses of operators. In order to do so, we first find a sufficient condition on general operators and use it to reduce the problem to whether the Berezin transform is one-to-one on related subclasses.

Original languageEnglish
Pages (from-to)3135-3166
Number of pages32
JournalJournal of Functional Analysis
Volume257
Issue number10
DOIs
Publication statusPublished - 2009 Nov 15

Keywords

  • Berezin transform
  • Half-space
  • Harmonic Bergman space
  • Toeplitz operator

ASJC Scopus subject areas

  • Analysis

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