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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Berezin regularity of domains in $\mathbb {C}^n$ and the essential norms of Toeplitz operators


Authors: Željko Čučković and Sönmez Şahutoğlu
Journal: Trans. Amer. Math. Soc. 374 (2021), 2521-2540
MSC (2020): Primary 47B35; Secondary 32W05
DOI: https://doi.org/10.1090/tran/8201
Published electronically: January 26, 2021
MathSciNet review: 4223024
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Abstract: For the open unit disc $\mathbb {D}$ in the complex plane, it is well known that if $\phi \in C(\overline {\mathbb {D}})$ then its Berezin transform $\widetilde {\phi }$ also belongs to $C(\overline {\mathbb {D}})$. We say that $\mathbb {D}$ is BC-regular. In this paper we study BC-regularity of some pseudoconvex domains in $\mathbb {C}^n$ and show that the boundary geometry plays an important role. We also establish a relationship between the essential norm of an operator in a natural Toeplitz subalgebra and its Berezin transform.


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Additional Information

Željko Čučković
Affiliation: Department of Mathematics & Statistics, University of Toledo, Toledo, Ohio 43606
MR Author ID: 294593
Email: Zeljko.Cuckovic@utoledo.edu

Sönmez Şahutoğlu
Affiliation: Department of Mathematics & Statistics, University of Toledo, Toledo, Ohio 43606
ORCID: 0000-0003-0490-0113
Email: Sonmez.Sahutoglu@utoledo.edu

Keywords: Berezin transform, Bergman kernel, $\overline {\partial }$-Neumann operator, convex domain, pseudoconvex domain
Received by editor(s): September 19, 2019
Received by editor(s) in revised form: April 1, 2020, and April 27, 2020
Published electronically: January 26, 2021
Article copyright: © Copyright 2021 American Mathematical Society