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 $L^2_a(\mathbb{D})$. More precisely, if ${\phi} = c g + d \bar{g}$, where $c,d\in\mathbb{C}$ and $g\in H^{\infty}(\mathbb{D})$, the space of all bounded analytic functions, then $T_{\phi}$ is invertible on $L^2_a(\mathbb{D})$ if and only if $\inf\limits_{z\in \mathbb{D}}\left|\widetilde{\,{\phi}}(z)\right|=\inf\limits_{z\in \mathbb{D}}|\phi(z)|>0$, where $\widetilde{\,{\phi}}$ is the Berezin transform of $\phi$.