In this paper we study the problem of the membership of H ∅ in the Hilbert-Schmidt class, when ∅ ∈ L ∞ (Ω) 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 Hφ̄ 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 Hφ̄ with φ holomorphic. © 2006 Birkhäuser Verlag Basel/Switzerland.
Raimondo, R. (2007). Hilbert-Schmidt Hankel Operators on the Bergman Space of Planar Domains. INTEGRAL EQUATIONS AND OPERATOR THEORY, 57, 425-449 [10.1007/s00020-006-1460-2].
Hilbert-Schmidt Hankel Operators on the Bergman Space of Planar Domains
RAIMONDO, ROBERTO
2007
Abstract
In this paper we study the problem of the membership of H ∅ in the Hilbert-Schmidt class, when ∅ ∈ L ∞ (Ω) 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 Hφ̄ 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 Hφ̄ with φ holomorphic. © 2006 Birkhäuser Verlag Basel/Switzerland.
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