# Toeplitz operators on pluriharmonic function spaces: Deformation quantization and spectral theory

## TL;DR

This paper investigates the asymptotic behavior and spectral properties of Toeplitz operators on pluriharmonic function spaces, establishing links to deformation quantization and providing insights into their essential spectra.

## Contribution

It introduces new asymptotic results for Toeplitz operators on pluriharmonic spaces and analyzes their spectral properties, extending the understanding of quantization in these contexts.

## Key findings

- Operator norms converge to supremum norms
- Product of Toeplitz operators approximates Toeplitz of product
- Commutators approximate Poisson brackets

## Abstract

Quantization and spectral properties of Toeplitz operators acting on spaces of pluriharmonic functions over bounded symmetric domains and $\mathbb C^n$ are discussed. Results are presented on the asymptotics \begin{align*} \| T_f^\lambda\|_\lambda &\to \| f\|_\infty\\ \| T_f^\lambda T_g^\lambda - T_{fg}^\lambda\|_\lambda &\to 0\\ \| \frac{\lambda}{i} [T_f^\lambda, T_g^\lambda] - T_{\{f,g\}}^\lambda\|_\lambda &\to 0 \end{align*} for $\lambda \to \infty$, where the symbols $f$ and $g$ are from suitable function spaces. Further, results on the essential spectrum of such Toeplitz operators with certain symbols are derived.