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Essential positivity for Toeplitz operators on the Fock space

Robert Fulsche

Abstract

In this short note, we discuss essential positivity of Toeplitz operators on the Fock space, as motivated by a recent question of Perälä and Virtanen. We give a proper characterization of essential positivity in terms of limit operators. A conjectured characterization of essential positivity of Perälä and Virtanen is disproven when the assumption of radiality is dropped. Nevertheless, when the symbol of the Toeplitz operator is of vanishing mean oscillation, we show that the conjecture of Perälä and Virtanen holds true, even without radiality.

Essential positivity for Toeplitz operators on the Fock space

Abstract

In this short note, we discuss essential positivity of Toeplitz operators on the Fock space, as motivated by a recent question of Perälä and Virtanen. We give a proper characterization of essential positivity in terms of limit operators. A conjectured characterization of essential positivity of Perälä and Virtanen is disproven when the assumption of radiality is dropped. Nevertheless, when the symbol of the Toeplitz operator is of vanishing mean oscillation, we show that the conjecture of Perälä and Virtanen holds true, even without radiality.
Paper Structure (2 sections, 13 theorems, 37 equations)

This paper contains 2 sections, 13 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Essential positivity for Toeplitz operators on the Fock space

Key Result

Theorem 1

Let $\mu$ be a radial real-valued Borel measure on $\mathbb D$ such that $|\mu|$ is a Carleson measure for the Bergman space $A^2(\mathbb D)$. Suppose that the limit $L = \lim_{|z| \to 1} \tilde{\mu}(z)$ exists. Then, $T_\mu$ is essentially positive on $A^2(\mathbb D)$ if and only if $L \geq 0$.

Theorems & Definitions (20)

  • Theorem : Perala_Virtanen2023
  • Conjecture
  • Theorem : Isralowitz_Zhu2010
  • Lemma 2.1
  • Theorem 2.2: Fulsche2020
  • Theorem 2.3: Bauer_Isralowitz2012
  • Theorem 2.4: Fulsche_Hagger
  • Corollary 2.5
  • Lemma 2.6
  • proof
  • ...and 10 more