Abstract.
In this paper we study the problem of the membership of H ϕ 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 H φ 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.
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Raimondo, R.C. Hilbert-Schmidt Hankel Operators on the Bergman Space of Planar Domains. Integr. equ. oper. theory 57, 425–449 (2007). https://doi.org/10.1007/s00020-006-1460-2
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DOI: https://doi.org/10.1007/s00020-006-1460-2