Positive Berezin liminf does not imply essential positivity for radial Toeplitz operators on Bergman and Fock spaces

TL;DR

This paper demonstrates that the radial Berezin liminf condition does not reliably indicate the essential positivity of Toeplitz operators on Bergman and Fock spaces, disproving a conjecture and highlighting limitations of scalar asymptotic tests.

Contribution

It provides explicit counterexamples showing the failure of the radial Berezin liminf criterion to detect essential positivity, disproving the Per"al"a--Virtanen conjecture in all dimensions.

Findings

Counterexamples with positive Berezin liminf but negative essential spectrum

Disproof of the Per"al"a--Virtanen conjecture in Bergman and Fock spaces

Radial Berezin liminf criterion fails universally across dimensions

Abstract

We study whether essential positivity \[ \sigma_{\mathrm{ess}}(T_f)\subset [0,\infty) \] of a radial Toeplitz operator on Bergman and Fock spaces can be detected from the asymptotic behavior of its Berezin transform. For bounded real-valued radial symbols on $A^2(\mathbb{D})$, Per\"al\"a and Virtanen conjectured that \[ \liminf_{|z|\to 1^-} \widetilde{f}(z)\ge 0 \] should be equivalent to essential positivity, and they asked the analogous question on Fock space. Such a criterion would turn a spectral question into a scalar asymptotic test. We prove that this fails even in the radial setting in which the conjecture was formulated. For every complex dimension $d\ge 1$, we construct explicit bounded real-valued radial symbols on the Bergman spaces $A^2(\mathbb{B}_d)$ and the Fock spaces $F^2(\mathbb{C}^d)$ whose Berezin transform has strictly positive limit inferior at the boundary (respectively, at infinity), while the essential spectrum of the corresponding Toeplitz operator contains a negative point. In particular, this disproves the Per\"al\"a--Virtanen conjecture in its original one-dimensional Bergman form and answers the analogous Fock-space question negatively; more generally, the radial Berezin liminf criterion fails in all dimensions in both settings. The underlying reason is that, for radial symbols, the Toeplitz eigenvalue sequence and the Berezin transform are different asymptotic averages of the same oscillatory symbol, and these averages damp the oscillation by different amounts.