The commutants of certain Toeplitz operators on weighted Bergman spaces

TL;DR

This paper characterizes functions whose Toeplitz operators commute with specific monomial or radial-type Toeplitz operators on weighted Bergman spaces in several complex variables.

Contribution

It provides a complete description of commuting functions for Toeplitz operators with monomial and radial symbols on weighted Bergman spaces.

Findings

Characterization of functions commuting with monomial Toeplitz operators

Description of functions commuting with radial-type Toeplitz operators

Extension of commutant results to several complex variables

Abstract

For $\alpha>-1$, let $A^2_{\alpha}$ be the corresponding weighted Bergman space of the unit ball in $\mathbb{C}^n$. For a bounded measurable function $f$, let $T_f$ be the Toeplitz operator with symbol $f$ on $A^2_{\alpha}$. This paper describes all the functions $f$ for which $T_f$ commutes with a given $T_g$, where $g(z)=z_{1}^{L_1}... z_{n}^{L_n}$ for strictly positive integers $L_1,..., L_n$, or $g(z)=|z_1|^{s_1}... |z_n|^{s_n}h(|z|)$ for non-negative real numbers $s_1,..., s_n$ and a bounded measurable function $h$ on $[0,1)$.