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

Abstract

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