On the Berger-Coburn phenomenon

In their previous work, the authors proved the Berger-Coburn phenomenon for compact and Schatten $S_p$ class Hankel operators $H_f$ on generalized Fock spaces when $1<p<\infty$, that is, for a bounded symbol $f$, if $H_f$ is a compact or Schatten class operator, then so is $H_{\bar f}$. More recently J.~Xia has provided a simple example that shows that there is no Berger-Coburn phenomenon for trace class Hankel operators on the classical Fock space $F^2$. Using Xia's example, we show that there is no Berger-Coburn phenomena for Schatten $S_p$ class Hankel operators on generalized Fock spaces $F^2_\varphi$ for any $0<p\le 1$. Our approach is based on the characterization of Schatten class Hankel operators while Xia's approach is elementary and heavily uses the explicit basis vectors of $F^2$, which cannot be found for the weighted Fock spaces that we consider. We also formulate four open problems.