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

TL;DR

This paper characterizes the compactness of Toeplitz operators with continuous symbols on weighted Bergman spaces over bounded pseudoconvex domains in complex space, linking it to boundary vanishing of the symbol.

Contribution

It provides a complete characterization of when Toeplitz operators are compact on these spaces, extending known results to more general domains and forms.

Findings

Toeplitz operator $T_{}$ is compact iff $$ vanishes on the boundary.

Compactness of $T^{p,q}_{}$ is equivalent to $=0$ on $ar{}$.

Results hold for weighted Bergman spaces and $ar{}$-closed forms.

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 $\Omega$.