Moment maps of Abelian groups and commuting Toeplitz operators acting on the unit ball

TL;DR

This paper constructs and analyzes commutative algebras of Toeplitz operators on the unit ball using moment maps of Abelian group actions, unifying and extending previous symbol sets with explicit spectral formulas.

Contribution

It introduces a general method to generate symbols for Toeplitz operators from moment maps of Abelian subgroups, encompassing known and new symbol sets with explicit spectral formulas.

Findings

Constructed symbol sets from moment maps produce commutative Toeplitz operator algebras.

Unified existing symbol sets and introduced new examples with explicit formulas.

Derived simplified spectral integral formulas for these Toeplitz operators.

Abstract

We prove that to every connected Abelian subgroup $H$ of the biholomorphisms of the unit ball $\mathbb{B}^n$ we can associate a set of bounded symbols whose corresponding Toeplitz operators generate a commutative $C^*$-algebra on every weighted Bergman space. These symbols are of the form $a(z) = f(\mu^H(z))$, where $\mu^H$ is the moment map for the action of $H$ on $\mathbb{B}^n$. We show that, for this construction, if $H$ is a maximal Abelian subgroup, then the symbols introduced are precisely the $H$-invariant symbols. We provide the explicit computation of moment maps to obtain special sets of symbols described in terms of coordinates. In particular, it is proved that our symbol sets have as particular cases all symbol sets from the current literature that yield Toeplitz operators generating commutative $C^*$-algebras on all weighted Bergman spaces on the unit ball $\mathbb{B}^n$. Furthermore, we exhibit examples that show that some of the symbol sets introduced in this work have not been considered before. Finally, several explicit formulas for the corresponding spectra of the Toeplitz operators are presented. These include spectral integral expressions that simplify the known formulas for maximal Abelian subgroups for the unit ball.