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

TL;DR

This work advances the invertibility theory for Toeplitz operators on the Bergman space by replacing positivity with a geometric condition ${\rm Re}\,\varphi \ge |{\rm Im}\,\varphi|^2$ and proving that $T_\varphi$ is invertible iff the Berezin transform $B(|\varphi|)$ is invertible in $L^\infty$. It further develops a detailed invertibility criterion for a broad class of harmonic polynomials, providing explicit (necessary and) sufficient conditions and thereby contributing to the Douglas problem in the Bergman setting. The paper also offers robust sufficient-condition frameworks, Poisson-extension reformulations for harmonic symbols, and iterated Berezin-transform results, complemented by illustrative examples and counterexamples. Collectively, these results sharpen the understanding of how symbol geometry and Berezin transforms govern invertibility and Fredholm properties in Bergman spaces, with implications for a wide range of analytic function spaces and symbol classes.

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.