Commutative algebras of Toeplitz operators and Lagrangian foliations

TL;DR

This paper explores the geometric structure underlying commutative algebras of Toeplitz operators on homogeneous bounded domains, revealing a connection to Lagrangian foliations when the algebra is sufficiently rich.

Contribution

It establishes that a sufficiently rich set of symbols defining Toeplitz operators induces a Lagrangian foliation structure on the domain.

Findings

Commutative Toeplitz operator algebras relate to Lagrangian foliations.

The geometric structure emerges when the algebra of symbols is sufficiently rich.

Provides a link between operator theory and symplectic geometry.

Abstract

Let $D$ be a homogeneous bounded domain of $\mathbb{C}^n$ and $\mathcal{A}$ a set of (anti--Wick) symbols that defines a commutative algebra of Toeplitz operators on every weighted Bergman space of $D$. We prove that if $\mathcal{A}$ is rich enough, then it has an underlying geometric structure given by a Lagrangian foliation.