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The Berezin liminf criterion fails for radial Toeplitz operators

Sam Looi

Abstract

We show that a positive limit inferior of the Berezin transform does not imply essential positivity for radial Toeplitz operators on the Bergman spaces $A^2(\mathbb B_d)$ and the Fock spaces $F^2(\mathbb C^d)$ in every complex dimension $d\ge 1$. In particular, this disproves the Perälä--Virtanen conjecture in its original one-dimensional Bergman form and shows that the corresponding radial Berezin criterion fails in all dimensions in both the Bergman and Fock settings. For each $d\ge 1$ we construct explicit bounded real-valued radial symbols $f$ for which $\liminf_{|z|\to1^-}\widetilde f(z)>0$ on $A^2(\mathbb B_d)$ and $\liminf_{|z|\to\infty}\widetilde f(z)>0$ on $F^2(\mathbb C^d)$, while the essential spectrum of $T_f$ contains a negative point. In both settings, the proofs reduce to explicit asymptotics for one-dimensional oscillatory integrals. The examples arise because, for radial symbols, the eigenvalue sequence and the Berezin transform are different asymptotic averages of the same oscillatory symbol, and these averages attenuate the oscillation by different factors.

The Berezin liminf criterion fails for radial Toeplitz operators

Abstract

We show that a positive limit inferior of the Berezin transform does not imply essential positivity for radial Toeplitz operators on the Bergman spaces and the Fock spaces in every complex dimension . In particular, this disproves the Perälä--Virtanen conjecture in its original one-dimensional Bergman form and shows that the corresponding radial Berezin criterion fails in all dimensions in both the Bergman and Fock settings. For each we construct explicit bounded real-valued radial symbols for which on and on , while the essential spectrum of contains a negative point. In both settings, the proofs reduce to explicit asymptotics for one-dimensional oscillatory integrals. The examples arise because, for radial symbols, the eigenvalue sequence and the Berezin transform are different asymptotic averages of the same oscillatory symbol, and these averages attenuate the oscillation by different factors.
Paper Structure (8 sections, 11 theorems, 207 equations)

This paper contains 8 sections, 11 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Fock space
  3. Radial Toeplitz operators on the Fock space
  4. A radial counterexample on the Fock space
  5. Bergman space
  6. Radial Toeplitz operators on the Bergman space
  7. A radial counterexample on the Bergman space
  8. Higher-dimensional variants

Key Result

Lemma 2.1

Let $f(z)=\varphi(|z|)$ be a bounded radial symbol. Then $T_f$ is diagonal with respect to the basis $(e_n)_{n\ge 0}$, and

Theorems & Definitions (27)

  • Conjecture 1.1: Perälä--Virtanen PeralaVirtanen
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • proof : Proof of Theorem \ref{['thm:fock-counterexample']}
  • Remark 2.5
  • ...and 17 more