Hilbert-Schmidt Hankel operators on the Segal-Bargmann space

Abstract

This paper considers Hankel operators on the Segal-Bargmann space of holomorphic functions on C n \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 ∈ L ∞ ( C n ) g \in L^{\infty }(\mathbb {C}^n) the Hankel operator H g H_g is of the Hilbert-Schmidt class if and only if H g ¯ 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 H_g and H g ¯ H_{\bar {g}} coincide. But, in general, if we deal with bounded symbols, only the inequality ‖ H g ‖ H S ≤ 2 ‖ H g ¯ ‖ H S \|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 of H g H_g implies the compactness of H g ¯ H_{\bar {g}} .