A local weighted Axler-Zheng theorem in ${\mathbb{C}}^n$

Abstract

The well-known Axler-Zheng theorem characterizes compactness of finite sums of finite products of Toeplitz operators on the unit disk in terms of the Berezin transform of these operators. Subsequently this theorem was generalized to other domains and appeared in different forms, including domains in ${\mathbb{C}}^n$ on which the $\overline{\partial }$-Neumann operator $N$ is compact. In this work we remove the assumption on $N$, and we study weighted Bergman spaces on smooth bounded pseudoconvex domains. We prove a local version of the Axler-Zheng theorem characterizing compactness of Toeplitz operators in the algebra generated by symbols continuous up to the boundary in terms of the behavior of the Berezin transform at strongly pseudoconvex points. We employ a Forelli-Rudin type inflation method to handle the weights.

arXiv commentMinor editorial changes, to appear in Pacific J. Math

Cited by in corpus (6)

• Convexity of the Berezin Range
• A sufficient condition for $L^p$ regularity of the Berezin transform
• Compactness of Toeplitz operators with continuous symbols on pseudoconvex domains in ${\mathbb{C}}^n$
• Compactness of Hankel and Toeplitz operators on convex Reinhardt domains in ${\mathbb{C}}^2$
• On compactness of products of Toeplitz operators
• On convergence of the Berezin transforms