Toeplitz operators on the weighted Bergman spaces of quotient domains

TL;DR

This paper explores Toeplitz operators on weighted Bergman spaces of quotient domains formed by finite pseudoreflection groups, using invariant and representation theory to analyze algebraic properties and specific operator problems.

Contribution

It introduces new identities and techniques for studying Toeplitz operators on quotient domains, including solutions to the zero-product problem and characterization of commuting pairs.

Findings

Derived identities for Toeplitz operators on quotient domains.

Solved the generalized zero-product problem.

Characterized commuting pairs of Toeplitz operators.

Abstract

Let $G$ be a finite pseudoreflection group and $\Omega\subseteq \mathbb C^d$ be a bounded domain which is a $G$-space. We establish identities involving Toeplitz operators on the weighted Bergman spaces of $\Omega$ and $\Omega/G$ using invariant theory and representation theory of $G.$ This, in turn, provides techniques to study algebraic properties of Toeplitz operators on the weighted Bergman space on $\Omega/G.$ We specialize on the generalized zero-product problem and characterization of commuting pairs of Toeplitz operators. As a consequence, more intricate results on Toeplitz operators on the weighted Bergman spaces on some specific quotient domains (namely symmetrized polydisc, monomial polyhedron, Rudin's domain) have been obtained.