Perälä–Virtanen conjecture on essential positivity of radial Bergman Toeplitz operators

Let $A^2(\mathbb D)$ be the Bergman space of the unit disk, let $f\in L^\infty(\mathbb D)$ be real-valued and radial, and let $T_f:A^2(\mathbb D)\to A^2(\mathbb D)$ be the Toeplitz operator with symbol $f$. Write $\widetilde f$ for its Berezin transform, and call $T_f$ essentially positive when

$$\sigma_{\mathrm{ess}}(T_f)\subset[0,\infty).$$

Perälä–Virtanen conjecture. The operator $T_f$ is essentially positive if and only if

$$\liminf_{|z|\to1^-}\widetilde f(z)\ge0.$$

The conjecture proposes that the boundary lower limit of the Berezin transform detects essential positivity for bounded real-valued radial symbols. The paper disproves it by constructing such symbols whose Berezin transform has strictly positive boundary liminf while the essential spectrum of the Toeplitz operator contains a negative point; the analogous Fock-space question is also answered negatively.