Vol. 294, No. 1, 2018

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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
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
Primary: 47B35
Secondary: 32W05
Milestones
Received: 23 April 2017
Revised: 25 September 2017
Accepted: 25 September 2017
Published: 5 January 2018
Authors
 Željko Čučković Department of Mathematics and Statistics University of Toledo Toledo, OH United States Sönmez Şahutoğlu Department of Mathematics and Statistics University of Toledo Toledo, OH United States Sabancı University Tuzla Istanbul Turkey Yunus E. Zeytuncu Department of Mathematics and Statistics University of Michigan-Dearborn Dearborn, MI United States