Algebraic properties of Toeplitz operators on generalized Fock spaces on $\mathbb{C}^d$

TL;DR

This paper investigates the algebraic structure of Toeplitz operators on generalized Fock spaces with specific weights, focusing on their commutants and zero-product conditions, revealing new insights into their operator algebra.

Contribution

It characterizes the commutant of Toeplitz operators with radial symbols and analyzes the zero-product problem for such operators on generalized Fock spaces.

Findings

Identified the commutant of Toeplitz operators with radial symbols.

Solved the zero-product problem for Toeplitz operators with radial functions.

Provided conditions under which the product of two Toeplitz operators is zero.

Abstract

We study two problems involving algebraic properties of Toeplitz operators on generalized Fock spaces on $\mathbb{C}^d$ with weights of the form $\left|z\right|^{2s} e^{-\left|z\right|^{2m}}$, $m\geq 1,\ s\geq 0$. We determine the commutant of a given Toeplitz operator with a radial symbol which satisfies certain growth conditions. We also discuss the equation $T_fT_g=0$, when $f$ or $g$ is radial.