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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Vydáno v:Advances in mathematics (New York. 1965) Ročník 404; s. 108411
Hlavní autoři: Le, Trieu, Tikaradze, Akaki
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier Inc 06.08.2022
Témata:
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
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Popis
Shrnutí: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