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Published in:
2024 | OriginalPaper | Chapter
On the Berger-Coburn Phenomenon
Authors : Zhangjian Hu, Jani A. Virtanen
Published in: Operator and Matrix Theory, Function Spaces, and Applications
Publisher: Springer Nature Switzerland
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Abstract
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.