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

Trevor Camper, Mishko Mitkovski

TL;DR

This work develops Szegő-type limit theorems for Toeplitz operators on a broad abstract RKHS framework and then specializes to concrete spaces. It combines an extended Berezin-Lieb inequality with a convergence analysis for the Berezin transform to prove limits of trace functionals of convex functions of Toeplitz operators, yielding explicit integral limits against the symbol. The results unify Bergman-space Szegő theory with an abstract, group-agnostic approach, and apply to weighted Bergman spaces, Segal-Bargmann-Fock, Paley-Wiener spaces, the classical Szegő setting, and both compact and locally compact Abelian groups via Følner sequences. The methodology offers a versatile bridge between operator-theoretic Szegő limits and harmonic analysis on groups, enabling new limit results and generalizations across several canonical spaces. Overall, the paper broadens the reach of Szegő-type asymptotics beyond classical settings to a cohesive, highly general framework with rich applications.

Abstract

We obtain Szegő-type limit theorems for Toeplitz operators on the weighted Bergman spaces $A^{2}_α(\mathbb{D})$, and on $L^{2}(G)$ where $G$ is a compact Abelian group. We also derive several abstract Szegő limit theorems which include many related classical Szegő limit theorems as a special case.

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

TL;DR

This work develops Szegő-type limit theorems for Toeplitz operators on a broad abstract RKHS framework and then specializes to concrete spaces. It combines an extended Berezin-Lieb inequality with a convergence analysis for the Berezin transform to prove limits of trace functionals of convex functions of Toeplitz operators, yielding explicit integral limits against the symbol. The results unify Bergman-space Szegő theory with an abstract, group-agnostic approach, and apply to weighted Bergman spaces, Segal-Bargmann-Fock, Paley-Wiener spaces, the classical Szegő setting, and both compact and locally compact Abelian groups via Følner sequences. The methodology offers a versatile bridge between operator-theoretic Szegő limits and harmonic analysis on groups, enabling new limit results and generalizations across several canonical spaces. Overall, the paper broadens the reach of Szegő-type asymptotics beyond classical settings to a cohesive, highly general framework with rich applications.

Abstract

We obtain Szegő-type limit theorems for Toeplitz operators on the weighted Bergman spaces , and on where is a compact Abelian group. We also derive several abstract Szegő limit theorems which include many related classical Szegő limit theorems as a special case.

Paper Structure

This paper contains 10 sections, 18 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Extended Berezin-Lieb Inequality
  3. Szegő Limit Theorems
  4. Convergence of the Berezin Transform
  5. Applications
  6. Bergman Space
  7. Segal-Bargmann-Fock Space
  8. Paley-Wiener Space
  9. The Classical Szegő Limit Theorem
  10. Locally Compact Abelian Groups

Key Result

Theorem 1.1

Let $\sigma:\mathbb{D}\to \mathbb{R}$ be a non-negative and bounded function such that $\sigma\in L^{1}(\mathbb{D},(1-|z|^{2})^{-2}dA(z))$. Then, for every continuous function $\psi: [0, \left\lVert \sigma \right\rVert_{\infty}]\to \mathbb{R}$ we have

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Proposition 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 21 more