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

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^λ\|_λ&\to \| f\|_\infty\\ \| T_f^λT_g^λ- T_{fg}^λ\|_λ&\to 0\\ \| \fracλ{i} [T_f^λ, T_g^λ] - T_{\{f,g\}}^λ\|_λ&\to 0 \end{align*} for $λ\to \infty$, 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.