Super Toeplitz operators on line bundles

Abstract

Let Lk be a high power of a hermitian holomorphic line bundle over a complex manifold X. Given a differential form f on X, we define a super Toeplitz operator Tf acting on the space of harmonic (0, q)-forms with values in Lk, with symbol f. The asymptotic distribution of its eigenvalues, when k tends to infinity, is obtained in terms of the symbol of the operator and the curvature of the line bundle L, given certain conditions on the curvature. For example, already when q = 0, i.e., the case of holomorphic sections, this generalizes a result of Bautet de Monvel and Guillemin to semi-positive line bundles. The asympotics are obtained from the asymptotics of the Bergman kernels of the corresponding harmonic spaces, which have independent interest. Applications to sampling are also given.