Equivariant asymptotics for Toeplitz operators

TL;DR

This paper extends the Tian-Zelditch asymptotic expansion to spectral projectors of Toeplitz operators, providing new insights into their asymptotic behavior in geometric quantization contexts.

Contribution

It introduces asymptotic analysis for spectral projectors of Toeplitz operators with shrinking spectral bands, generalizing previous equivariant kernel expansions.

Findings

Derived asymptotic formulas for spectral projectors of Toeplitz operators.

Extended geometric quantization techniques to non-complete linear series.

Provided new tools for analyzing spectral bounds in complex geometry.

Abstract

In recent years, the Tian-Zelditch asymptotic expansion for the equivariant components of the Szeg\"{o} kernel of a polarized complex projective manifold, and its subsequent generalizations in terms of scaling limits, have played an important role in algebraic, symplectic, and differential geometry. A natural question is whether there exist generalizations in which the projector onto the spaces of holomorphic sections can be replaced by the projector onto more general (non-complete) linear series. One case that lends itself to such analysis, and which is natural from the point of view of geometric quantization, is given by the linear series determined by imposing spectral bounds on an invariant self-adjoint Toeplitz operator. In this paper we focus on the asymptotics of the spectral projectors associated to slowly shrinking spectral bands.