Toeplitz operators with quasi-radial quasi-homogeneous symbols and bundles of Lagrangian frames

TL;DR

This paper demonstrates that quasi-homogeneous symbols on complex projective space generate commutative Toeplitz operator algebras on weighted Bergman spaces, extending known results from the unit ball and linking algebraic structures to geometric properties.

Contribution

It extends the theory of Toeplitz operators with quasi-homogeneous symbols to projective space, establishing commutativity and geometric connections in this compact setting.

Findings

Quasi-homogeneous symbols produce commutative Toeplitz algebras on projective space.

These algebras are Banach but not $C^*$-algebras.

A strong link between symbols, algebras, and the geometry of $ ext{P}^n( ext{C})$ is established.

Abstract

We prove that the quasi-homogenous symbols on the projective space $\mathbb{P}^n(\mathbb{C})$ yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball $\mathbb{B}^n$. These algebras are Banach but not $C^*$. We prove the existence of a strong link between such symbols and algebras with the geometry of $\mathbb{P}^n(\mathbb{C})$. The latter is also proved for the corresponding symbols and algebras on $\mathbb{B}^n$.