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

Željko Čučković, Sönmez Şahutoğlu and Yunus E. Zeytuncu

Vol. 294 (2018), No. 1, 89–106

DOI: 10.2140/pjm.2018.294.89

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 ${ℂ}^n$ 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.

Keywords

Axler–Zheng theorem, Toeplitz operators, pseudoconvex domains

Mathematical Subject Classification 2010

Primary: 47B35

Secondary: 32W05

Milestones

Received: 23 April 2017

Revised: 25 September 2017

Accepted: 25 September 2017

Published: 5 January 2018