Schatten class Hankel operators on the Segal-Bargmann space and the Berger-Coburn phenomenon

Abstract

We characterize Schatten $p$-class Hankel operators $H_f$ on the Segal-Bargmann space when $0<p<\infty$ in terms of our recently introduced notion of integral distance to analytic functions in $\mathbb{C}^n$. Our work completes the study inspired by a theorem of Berger and Coburn on compactness of Hankel operators and subsequently initiated twenty years ago by Xia and Zheng, who obtained a characterization of the simultaneous membership of $H_f$ and $H_{\overline f}$ in Schatten classes $S_p$ when $1\le p<\infty$ in terms of the standard deviation of $f$. As an application, we give a positive answer to their question of whether $H_f\in S_p$ implies $H_{\overline{f}}\in S_p$ when $f\in L^\infty$ and $1<p<\infty$, which was previously solved for $p=2$ and $n=1$ by Xia and Zheng and for $p=2$ in any dimension by Bauer in 2004. In addition, we prove our results in the context of weighted Segal-Bargmann spaces, which include the standard and Fock-Sobolev weights