A generalization of the Brown–Halmos theorems for the unit ball

In this paper we generalize the classical theorems of Brown and Halmos about algebraic properties of Toeplitz operators to Bergman spaces over the unit ball in several complex variables. A key result, which is of independent interest, is the characterization of summable functions u on the unit ball...

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Bibliographic Details
Published in:	Advances in mathematics (New York. 1965) Vol. 404; p. 108411
Main Authors:	Le, Trieu, Tikaradze, Akaki
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 06.08.2022
Subjects:	Bergman space Brown-Halmos theorems Toeplitz operators Zero-product problems Bergman space 32A36 Zero-product problems Toeplitz operators Brown-Halmos theorems 47B35
ISSN:	0001-8708, 1090-2082
Online Access:	Get full text
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Summary:	In this paper we generalize the classical theorems of Brown and Halmos about algebraic properties of Toeplitz operators to Bergman spaces over the unit ball in several complex variables. A key result, which is of independent interest, is the characterization of summable functions u on the unit ball whose Berezin transform can be written as a finite sum ∑jfjg¯j with all fj,gj being holomorphic. In particular, we show that such a function must be pluriharmonic if it is sufficiently smooth and bounded. We also settle an open question about M-harmonic functions. Our proofs employ techniques and results from function and operator theory as well as partial differential equations.
ISSN:	0001-8708 1090-2082
DOI:	10.1016/j.aim.2022.108411