Local Asymptotics for Slowly Shrinking Spectral Bands of a Berezin–Toeplitz Operator

Motivated by geometric quantization, we study how the asymptotics of certain spectral projectors of an invariant self-adjoint Toeplitz operator relate to the full level-k Szegö kernel associated to a positive line bundle on a compact symplectic manifold. These operators quantize classical observables on the "phase space" M, and the semiclassical regime corresponds to k →+∞. We focus on the asymptotics of the spectral projectors of slowly shrinking spectral bands. If "energy" is a zeroth-order Toeplitz operator, then for ξ <1/2, spectral bands of width O (k-ξ) centered at E asymptotically capture all of the Szegö kernel at points of classical energy E. On the same classical locus, these bands also realize a version of Tian's almost isometry. © 2010 The Author. Published by Oxford University Press. All rights reserved.