Commutants of the sum of two quasihomogeneous Toeplitz operators

TL;DR

This paper characterizes the commutants of sums of two quasihomogeneous Toeplitz operators on the Bergman space, extending previous results to more general radial functions.

Contribution

It generalizes existing commutation results by replacing specific radial functions with arbitrary powers of r, broadening the class of Toeplitz operators analyzed.

Findings

Commutants are scalar multiples of the original operator under generalized conditions.

The result holds for a wider class of radial functions beyond the previously studied forms.

Provides a complete characterization of commuting Toeplitz operators with generalized radial symbols.

Abstract

A major open question in the theory of Toeplitz operator on the Bergman space of the unit disk of the complex plane is the complete characterization of the set of all Toeplitz operators that commute with a given operator. In \cite{al}, the authors showed that when a sum $S=T_{e^{im\theta}f}+T_{e^{il\theta}g}$, where $f$ and $g$ are radial functions, commutes with a sum $T=T_{e^{ip\theta}r^{(2M+1)p}}+T_{e^{is\theta}r^{(2N+1)s}}$, then $S$ must be of the form $S=cT$, where $c$ is a constant. In this article, we will replace $r^{(2M+1)p}$ and $r^{(2N+1)s}$ with $r^n$ and $r^d$, where $n$ and $d$ are in $\mathbb{N}$, and we will show that the same result holds.