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

Abstract

We establish an abstract Szegő-type limit theorem for Toeplitz operators associated with normalized continuous Parseval frames. Our assumptions consist essentially of a natural normalization of the frame measures and an approximate-identity condition for the normalized frame vectors. Under these hypotheses, the Szegő asymptotics follow from a general argument based on Berezin--Lieb inequalities and convergence of Berezin transforms, independently of the particular geometry of the underlying space. As a principal application, we obtain a Szegő-type limit theorem for Toeplitz operators on weighted Bergman spaces of the unit ball. This setting lies outside the standard Widom-type localization approach because the underlying hyperbolic measure has exponential volume growth. We also obtain Szegő limit theorems for metrizable compact and locally compact Abelian groups, extending the classical compact-group results to more general symbols and recovering recent results for multidimensional tori as special cases. Further applications include Gabor localization operators, Paley--Wiener spaces, eigenvalue distribution, and sampling density results.

v4: 35 pages. As compared to previous versions, v4 presents improved results and exposition to v3. 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