A geometric condition for the invertibility of Toeplitz operators on the Bergman space

TL;DR

This paper introduces a new geometric condition for the invertibility of Toeplitz operators on the Bergman space, linking it to the invertibility of the Berezin transform, and explores related harmonic polynomial cases.

Contribution

It proposes a novel geometric criterion for invertibility that relaxes positivity assumptions and characterizes invertibility via the Berezin transform.

Findings

Invertibility of Toeplitz operators is characterized by the Berezin transform under the new geometric condition.

The paper extends the analysis to harmonic polynomials and their associated Toeplitz operators.

Several examples and related results are provided to illustrate the theory.

Abstract

Invertibility of Toeplitz operators on the Bergman space and the related Douglas problem are long standing open problems. In this paper we study the invertibility problem under the novel geometric condition on the image of the symbols, which relaxes the standard positivity condition. We show that under our geometric assumption, the Toeplitz operator $T_\varphi$ is invertible if and only if the Berezin transform of $|\varphi|$ is invertible in $L^{\infty}$. It is well known that the Douglas problem is still open for harmonic functions. We study a class of rather general harmonic polynomials and characterize the invertibility of the corresponding Toeplitz operators. We also give a number of related results and examples.