On the Weyl law for Toeplitz operators

Article type: Research Article

Authors: Paoletti, Roberto

Affiliations: Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Milano, Italy

Note: [] Address for correspondence: Roberto Paoletti, Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via R. Cozzi 53, 20125 Milano, Italy. E-mail: roberto.paoletti@unimib.it.

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). The global Weyl law follows by integration.

Keywords: Toeplitz operators, Weyl law, positive line bundle, Szegö kernel, Fourier integral operator

DOI: 10.3233/ASY-2008-0929

Journal: Asymptotic Analysis, vol. 63, no. 1-2, pp. 85-99, 2009