On the Weyl law for Toeplitz operators

TL;DR

This paper revisits the Weyl law for Toeplitz operators in the context of positive line bundles, employing microlocal analysis and Szego kernel techniques to derive local spectral asymptotics.

Contribution

It introduces a new local Weyl law for Toeplitz operators using microlocal methods and spectral function estimates, enhancing previous global results.

Findings

Established a local Weyl law for Toeplitz operators

Connected microlocal analysis with spectral function estimates

Provided pointwise spectral asymptotics in the positive line bundle setting

Abstract

A Weyl law for Toeplitz operators was proved by Boutet de Monvel and Guillemin for general Toeplitz structures. In the setting of positive line bundles, we revisit this theme in light of local asymptotic techniques based on the microlocal theory of the Szego kernel. By pairing this approach with classical arguments used to estimate the spectral function of a pseudodifferential operator, we first establish a local Weyl law (that is, a pointwise estimate on the spectral function of the Toeplitz operator).