Hilbert-Schmidt Hankel operators on the Segal-Bargmann space

This paper considers Hankel operators on the Segal-Bargmann space of holomorphic functions on $\mathbb{C}^n$ that are square integrable with respect to the Gaussian measure. It is shown that in the case of a bounded symbol $g \in L^{\infty}(\mathbb{C}^n)$ the Hankel operator $H_g$ is of the Hilbert-Schmidt class if and only if $H_{\bar{g}}$ is Hilbert-Schmidt. In the case where the symbol is square integrable with respect to the Lebesgue measure it is known that the Hilbert-Schmidt norms of the Hankel operators $H_g$ and $H_{\bar{g}}$ coincide. But, in general, if we deal with bounded symbols, only the inequality $\Vert H_g\Vert _{HS}\leq2\Vert H_{\bar{g}}\Vert _{HS}$ can be proved. The results have a close connection with the well-known fact that for bounded symbols the compactness of $H_g$ implies the compactness of $H_{\bar{g}}$.