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

TL;DR

This paper extends the Brown-Halmos theorems to Bergman spaces on the unit ball in several complex variables, characterizing functions with specific Berezin transform properties and addressing open questions in harmonic analysis.

Contribution

It generalizes classical operator theorems to higher dimensions and characterizes functions with finite Berezin transform representations, including solutions to an open problem.

Findings

Characterization of functions with Berezin transform as finite sums of holomorphic products

Proof that sufficiently smooth bounded functions with such transforms are pluriharmonic

Resolution of an open question about -harmonic functions

Abstract

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 $\sum_{j}f_j\,\bar{g}_j$ with all $f_j, g_j$ 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 $\mathcal{M}$-harmonic functions. Our proofs employ techniques and results from function and operator theory as well as partial differential equations.