Compactness of Toeplitz operators with continuous symbols on pseudoconvex domains in ${\mathbb{C}}^n$

Abstract

Let $\hat{I}\text{c◯}$ be a bounded pseudoconvex domain in ${\mathbb{C}}^n$ with Lipschitz boundary and $\ddot{I}•$ be a continuous function on $\overline{\hat{I}}\text{c◯}$. We show that the Toeplitz operator $T_{\ddot{I}}•$ with symbol $\ddot{I}•$ is compact on the weighted Bergman space if and only if $\ddot{I}•$ vanishes on the boundary of $\hat{I}\text{c◯}$. We also show that compactness of the Toeplitz operator $T_{\ddot{I}}^{p,q}•$ on $\overline{\partial }$-closed $(p,q)$-forms for $0\le p\le n$ and $q\ge 1$ is equivalent to $\ddot{I}•=0$ on $\hat{I}\text{c◯}$.

Final version. Fixed one reference. To appear in Proc. Amer. Math. Soc