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

Abstract

In this paper we study the problem of the membership of H φ in the Hilbert-Schmidt class, when φ ∈ 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 Hϕ 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 Hϕ with ϕ holomorphic. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47B38.

Schatten-von Neumann Hankel Operators on the Bergman Space of Planar Domains

“…In fact this result together with the well-known fact that the reproducing kernel of the unit disk is given by w) it is possible to prove that (see [7])…”

“…Recently (see [7]) the author has found a characterization for the Hilbert-Schmidt case (i.e. p = 2).…”

“…With the symbols K Ωj (z, w), K Ω (z, w), K ∆ (z, w) we denote the Bergman kernel on Ω j , Ω, and ∆ respectively. If we define L p a (Ω 1 ) = {f ∈ L p (Ω)| f is holomorphic} and L p a (∆) = {f ∈ L p (∆)| f is holomorphic}, then we have the following (see [7]) Theorem 3.1. There exists an isomorphism I :…”

“…The next Lemma gives an useful decomposition of the space L 2 a (Ω), in fact we have (see [7]) (P j f )(z) + (P 0 f )(z)…”