# Journal of Operator Theory

Volume 75, Issue 2, Spring 2016 pp. 475-495.

Invertibility of Toeplitz operators via Berezin transforms

**Authors**:
Xianfeng Zhao (1) and Dechao Zheng (2)

**Author institution:**(1) College of Mathematics and Statistics,
Chongqing University, Chongqing, 401331, P.R. China (2) Center of
Mathematics, Chongqing University, Chongqing, 401331, P.R. China and
Department of Mathematics, Vanderbilt University, Nashville, TN 37240,
U.S.A.

**Summary: **We obtain a sufficient condition for a Toeplitz operator to be invertible
on the Bergman space via the $n$-th Berezin transforms of its symbol.
For a harmonic symbol, we obtain a sufficient condition for a Toeplitz operator
to be invertible on the Bergman space via only the Berezin transform of the symbol,
which is analogous to the Chang-Tolokonnikov-Nikolski conditions on the Hardy space.
For a nonnegative symbol, we prove that the Toeplitz operator is invertible on the Bergman
space if and only if its Berezin transform is bounded below by a fixed positive constant on the
unit disk.

**DOI: **http://dx.doi.org/10.7900/jot.2015jul07.2082

**Keywords: **Bergman space, invertible Toeplitz operators,
Berezin transform

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