Toeplitz operators and group-moment coordinates for quasi-elliptic and quasi-hyperbolic symbols

TL;DR

This paper develops spectral formulas for Toeplitz operators with specific invariant symbols on complex domains using symplectic geometry, demonstrating the effectiveness of geometric methods in operator analysis.

Contribution

It introduces a novel application of Hamiltonian actions and moment maps to diagonalize Toeplitz operators with quasi-elliptic and quasi-hyperbolic symbols.

Findings

Spectral integral formulas derived for Toeplitz operators.

Geometric methods simplify the analysis of invariant operators.

Demonstrates the power of symplectic geometry in operator theory.

Abstract

For $\mathbb{B}^n$ the $n$-dimensional unit ball and $D_n$ its Siegel unbounded realization, we consider Toeplitz operators acting on weighted Bergman spaces with symbols invariant under the actions of the maximal Abelian subgroups of biholomorphisms $\mathbb{T}^n$ (quasi-elliptic) and $\mathbb{T}^n \times \mathbb{R}_+$ (quasi-hyperbolic). Using geometric symplectic tools (Hamiltonian actions and moment maps) we obtain simple diagonalizing spectral integral formulas for such kinds of operators. Some consequences show how powerful the use of our differential geometric methods are.