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

We give a complete characterization of Schatten class Hankel operators H f H_f acting on weighted Segal-Bargmann spaces F 2 ( φ ) F^2(\varphi ) using the notion of integral distance to analytic functions in C n \mathbb {C}^n and Hörmander’s ∂ ¯ \bar \partial -theory. Using our characterization, for f ∈ L ∞ f\in L^\infty and 1 > p > ∞ 1>p>\infty , we prove that H f H_f is in the Schatten class S p S_p if and only if H f ¯ ∈ S p H_{\bar {f}}\in S_p , which was previously known only for the Hilbert-Schmidt class S 2 S_2 of the standard Segal-Bargmann space F 2 ( φ ) F^2(\varphi ) with φ ( z ) = α | z | 2 \varphi (z) = \alpha |z|^2 .