Hilbert-Schmidt Hankel Operators on the Bergman Space of Planar Domains

Abstract

In this paper we study the problem of the membership of Hϕ in the Hilbert-Schmidt class, when \( \phi \in L^{\infty } (\Omega) \) and Ω is a planar domain. We find a necessary and sufficient condition.We apply this result to the problem of joint membership of Hφ and \( H_{{\overline{\varphi } }} \) in the Hilbert-Schmidt class. Using the notion of Berezin Transform and a result of K. Zhu we are able to give a necessary and sufficient condition. Finally, we recover a result of Arazy, Fisher and Peetre on the case \( H_{{\overline{\varphi } }} \) with φ holomorphic.