A geometric condition for the invertibility of Toeplitz operators on the Bergman space
Volume 284 / 2025
Studia Mathematica 284 (2025), 147-164
MSC: Primary 47B35; Secondary 47B91
DOI: 10.4064/sm240829-16-5
Published online: 9 September 2025
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 a novel geometric condition on the images 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.
