Separately Radial and Radial Toeplitz Operators on the Projective Space and Representation Theory

TL;DR

This paper provides a new representation theoretic proof that Toeplitz operators with radial and separately radial symbols on complex projective space generate commutative $C^*$-algebras, and extends spectral formulas to intermediate groups.

Contribution

It introduces a simpler, representation-theoretic proof of commutativity and extends spectral formulas to groups between $\

Findings

The $C^*$-algebras generated by these Toeplitz operators are commutative.

The proof relies on the existence of multiplicity-free representations.

Spectral formulas are extended to intermediate groups between $\

Abstract

We consider separately radial (with corresponding group $\mathbb{T}^n$) and radial (with corresponding group $\mathrm{U}(n))$ symbols on the projective space $\mathbb{P}^n(\mathbb{C})$, as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the $C^*$-algebras generated by each family of such Toeplitz operators are commutative. We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the commutativity of the $C^*$-algebras is a consequence of the existence of multiplicity-free representations. Furthermore, our method shows how to extend the current formulas for the spectra of the corresponding Toeplitz operators to any closed group lying between $\mathbb{T}^n$ and $\mathrm{U}(n)$.