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

## TL;DR

This paper extends the Axler-Zheng theorem to weighted Bergman spaces on smooth bounded pseudoconvex domains in ^n, characterizing Toeplitz operator compactness locally at strongly pseudoconvex points without assuming the ar-Neumann operator is compact.

## Contribution

It establishes a local version of the Axler-Zheng theorem for weighted Bergman spaces on pseudoconvex domains, removing the previous compactness assumption on the ar-Neumann operator.

## Key findings

- Characterizes Toeplitz operator compactness via Berezin transform behavior.
- Develops a local theorem applicable at strongly pseudoconvex boundary points.
- Uses a Forelli-Rudin inflation method to handle weights.

## 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.