Essential positivity for Toeplitz operators on the Fock space

TL;DR

This paper characterizes the essential positivity of Toeplitz operators on the Fock space using limit operators, disproves a conjecture in the non-radial case, and confirms it for symbols with vanishing mean oscillation.

Contribution

It provides a new characterization of essential positivity via limit operators and clarifies the conditions under which a recent conjecture holds or fails.

Findings

Characterization of essential positivity in terms of limit operators

Disproof of the conjecture when radiality is not assumed

Validation of the conjecture for symbols with vanishing mean oscillation

Abstract

In this short note, we discuss essential positivity of Toeplitz operators on the Fock space, as motivated by a recent question of Per\"al\"a and Virtanen. We give a proper characterization of essential positivity in terms of limit operators. A conjectured characterization of essential positivity of Per\"al\"a 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 Per\"al\"a and Virtanen holds true, even without radiality.