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

Abstract

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^n$ with Lipschitz boundary and $\phi$ be a continuous function on $\overline{\Omega}$. We show that the Toeplitz operator $T_{\phi}$ with symbol $\phi$ is compact on the weighted Bergman space if and only if $\phi$ vanishes on the boundary of $\Omega$. We also show that compactness of the Toeplitz operator $T^{p,q}_{\phi}$ on $\overline{\partial}$-closed $(p,q)$-forms for $0\leq p\leq n$ and $q\geq 1$ is equivalent to $\phi=0$ on $\Omega$.