Commuting Toeplitz Operators on Bounded Symmetric Domains and Multiplicity-Free Restrictions of Holomorphic Discrete Series

TL;DR

This paper establishes the existence of commutative algebras generated by Toeplitz operators on weighted Bergman spaces over bounded symmetric domains, linking their symbols to subgroup invariance and multiplicity-free restrictions of holomorphic discrete series.

Contribution

It introduces a new framework connecting Toeplitz operator algebras with multiplicity-free restrictions of holomorphic discrete series on bounded symmetric domains.

Findings

Existence of commutative $C^*$-algebras generated by Toeplitz operators.

Characterization of invariant symbols under subgroup actions.

Complete description of subgroups yielding commuting Toeplitz operators in the compact case.

Abstract

For any given bounded symmetric domain, we prove the existence of commutative $C^*$-algebras generated by Toeplitz operators acting on any weighted Bergman space. The symbols of the Toeplitz operators that generate such algebras are defined by essentially bounded functions invariant under suitable subgroups of the group of biholomorphisms of the domain. These subgroups include the maximal compact groups of biholomorphisms. We prove the commutativity of the Toeplitz operators by considering the Bergman spaces as the underlying space of the holomorphic discrete series and then applying known multiplicity-free results for restrictions to certain subgroups of the holomorphic discrete series. In the compact case we completely characterize the subgroups that define invariant symbols that yield commuting Toeplitz operators in terms of the multiplicity-free property.