Hilbert-Schmidt Hankel operators on the Segal-Bargmann space

Abstract

This paper considers Hankel operators on the Segal-Bargmann space of holomorphic functions onCn\mathbb {C}^nthat are square integrable with respect to the Gaussian measure. It is shown that in the case of a bounded symbolg∈L∞(Cn)g \in L^{\infty }(\mathbb {C}^n)the Hankel operatorHgH_gis of the Hilbert-Schmidt class if and only ifHg¯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 operatorsHgH_gandHg¯H_{\bar {g}}coincide. But, in general, if we deal with bounded symbols, only the inequality‖Hg‖HS≤2‖Hg¯‖HS\|H_g\|_{HS}\leq 2\|H_{\bar {g}}\|_{HS}can be proved. The results have a close connection with the well-known fact that for bounded symbols the compactness ofHgH_gimplies the compactness ofHg¯H_{\bar {g}}.