The Restriction Principle and Commuting Families of Toeplitz Operators on the Unit Ball

TL;DR

This paper studies Toeplitz operators on the unit ball, proving their pairwise commutativity when associated with maximal abelian subgroups of a certain Lie group, by explicitly developing the restriction principle and Segal-Bargmann transform.

Contribution

It provides a detailed proof of commutativity for Toeplitz operators linked to maximal abelian subgroups, including explicit transforms and diagonalization methods.

Findings

Proves pairwise commutativity of Toeplitz operators with maximal abelian subgroup invariance.

Develops explicit restriction principles and Segal-Bargmann transforms for each subgroup.

Achieves explicit simultaneous diagonalization of these Toeplitz operators.

Abstract

On the unit ball B^n we consider the weighted Bergman spaces H_\lambda and their Toeplitz operators with bounded symbols. It is known from our previous work that if a closed subgroup H of \widetilde{\SU(n,1)} has a multiplicity-free restriction for the holomorphic discrete series of $\widetilde{\SU(n,1)}$, then the family of Toeplitz operators with H-invariant symbols pairwise commute. In this work we consider the case of maximal abelian subgroups of \widetilde{\SU(n,1)} and provide a detailed proof of the pairwise commutativity of the corresponding Toeplitz operators. To achieve this we explicitly develop the restriction principle for each (conjugacy class of) maximal abelian subgroup and obtain the corresponding Segal-Bargmann transform. In particular, we obtain a multiplicity one result for the restriction of the holomorphic discrete series to all maximal abelian subgroups. We also observe that the Segal-Bargman transform is (up to a unitary transformation) a convolution operator against a function that we write down explicitly for each case. This can be used to obtain the explicit simultaneous diagonalization of Toeplitz operators whose symbols are invariant by one of these maximal abelian subgroups.