The Brown-Halmos theorems on the Fock space

TL;DR

This paper extends the Brown-Halmos theorems to the Fock space, explores the Berezin transform's range, and addresses an open question about Toeplitz operator zero products, revealing increased complexity over classical spaces.

Contribution

It generalizes the Brown-Halmos theorems to Fock space and solves an open problem on Toeplitz operator zero products, highlighting new complexities.

Findings

Non-pluriharmonic functions can be expressed as finite sums involving holomorphic functions.

The range of the Berezin transform includes certain non-pluriharmonic functions.

The zero product problem for Toeplitz operators is resolved in the Fock space context.

Abstract

In this paper, we extend the Brown-Halmos theorems to the Fock space and investigate the range of the Berezin transform. We observe that there are non-pluriharmonic functions $u$ that can be written as a finite sum $B(u)=\sum_lf_l\overline{g_l}$, where $f_l,g_l$ are holomorphic functions belonging to the class $\mathrm{Sym}(\mathbb{C}^n)$. In addition, we solve an open question about the zero product of Toeplitz operators, which was posed by Bauer et al. in 2015. Our results reveal that the Brown-Halmos theorems on the Fock space are more complicated than that on the classical Bergman space.