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 $\varphi$ be a continuous function on $\overline{\Omega }$. We show that the Toeplitz operator $T_{\varphi }$ with symbol $\varphi$ is compact on the weighted Bergman space if and only if $\varphi$ vanishes on the boundary of $\Omega$. We also show that compactness of the Toeplitz operator $T_{\varphi }^{p,q}$ on $\overline{\partial }$-closed $(p,q)$-forms for $0\le p\le n$ and $q\ge 1$ is equivalent to $\varphi =0$ on $\Omega$.

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

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Cited by in corpus (4)

• Compactness of Toeplitz operators with continuous symbols on pseudoconvex domains in ${\mathbb{C}}^n$
• Compactness of composition operators on the Bergman space of the bidisc
• Compactness of Hankel and Toeplitz operators on convex Reinhardt domains in ${\mathbb{C}}^2$
• On compactness of products of Toeplitz operators