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^\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.