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Toeplitz Operators on Pluriharmonic Function Spaces: Deformation Quantization and Spectral Theory

  • Robert FulscheEmail author
Article
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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
$$\begin{aligned} \Vert T_f^\lambda \Vert _\lambda&\rightarrow \Vert f\Vert _\infty \\ \Vert T_f^\lambda T_g^\lambda - T_{fg}^\lambda \Vert _\lambda&\rightarrow 0\\ \Vert \frac{\lambda }{i} [T_f^\lambda , T_g^\lambda ] - T_{\{f,g\}}^\lambda \Vert _\lambda&\rightarrow 0 \end{aligned}$$
for \(\lambda \rightarrow \infty \), 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.

Keywords

Toeplitz operators Pluriharmonic functions Quantization Essential spectrum 

Mathematics Subject Classification

Primary 47B35 Secondary 30H20 47A53 81S10 

Notes

Acknowledgements

The author wants to thank Wolfram Bauer for his help and support and Raffael Hagger for valuable discussions. The author also appreciates the referees comments.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für AnalysisLeibniz Universität HannoverHannoverGermany

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