Invertibility of Bergman Toeplitz operators

Abstract

In this paper, we establish the invertibility of the Berezin transform of the symbol as a necessary and sufficient condition for the invertibility of the Toeplitz operator on the Bergman space $L_a^2(\mathbb{D})$. More precisely, if $\ddot{I}•=cg+d\bar{g}$, where $c,d\in \mathbb{C}$ and $g\in H^{\infty }(\mathbb{D})$, the space of all bounded analytic functions, then $T_{\ddot{I}}•$ is invertible on $L_a^2(\mathbb{D})$ if and only if $\underset{z\in \mathbb{D}}{\inf }\left|\widetilde{\ddot{I}•}(z)\right|=\underset{z\in \mathbb{D}}{\inf }|\ddot{I}•(z)|>0$, where $\widetilde{\ddot{I}•}$ is the Berezin transform of $\ddot{I}•$.

Preliminary version