A Semi-Classical Szegő-type Limit Theorem for Toeplitz Operators

Abstract

We obtain Szeg\H o-type limit theorems for Toeplitz operators on the weighted Bergman spaces $A_{\hat{I}}^2\pm ({\mathbb{B}}^n)$, and on $L^2(G)$, presenting separate formulations for compact and locally compact Abelian groups. Furthermore, we establish a broad class of abstract Szegő limit theorems that unify and extend many classical results.

v3: 23 pages. The v2 reflects an improvement to L^{1}(G) symbols in Theorem 1.2 as compared to v1. v3 contains improvements to the main results of v2, improved forms of the Weyl law, along with several new applications and corollaries of the results therein