Hankel operators in the von Neumann-Schatten classes
type: Item
instance of: scholarly article
title: Hankel operators in the von Neumann-Schatten classes
zbMATH Open document ID: 0618.47019
author: F. Holland
author: David O. Walsh
published in: Illinois Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: In his study of a Hankel operator on the Hardy space \(H^ 2\) of the disc D, Bonsall introduced the unit vectors \(v_ z(w)=\sqrt{1-| z| ^ 2}/(1-\bar zw)\) (w\(\in D)\) indexed by the points \(z\in D\) and proceeded to show that A is a bounded Hankel operator if and only if \(\{Av_ z:\) \(z\in D\}\) is a bounded subset of \(H^ 2.\) The purpose of this paper is to try to find conditions which relate the quantities \(\| Av_ z\|\) with the property that A belongs to the von Neumann-Schatten class \(C_ p\) \((1\leq p<\infty)\). A complete characterization is obtained only when \(p=2\). For other values of p implications in one direction only are obtained (necessary conditions that \(A\in C_ p\) for the most part) but it is shown that the converse implications do not hold. In deriving these results a necessary and sufficient condition that the analytic function f belongs to the Besov space \(B_ p^{1/p}\) is found in terms of the action of f on the group of Mobius transformations on D. This result is analogous to a well known characterization of the space BMOA. Bonsall also considered unit vectors \(u_ n(\zeta)\), \(n\geq 0\), \(\zeta\) \(\in \partial D\), the counterparts of the \(v_ z\) on the unit circle. Conditions in terms of the \(\| Au_ n(\zeta)\|\), completely analogous to those for \(\| Av_ u\|\), are obtained, including a necessary condition that \(A\in C_ 1\). Again the condition is shown to be not sufficient.
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 4002863
zbMATH Keywords: Hankel operator
zbMATH Keywords: Hardy space
zbMATH Keywords: von Neumann-Schatten class
zbMATH Keywords: Besov space
zbMATH Keywords: group of Möbius transformations
zbMATH Keywords: space BMOA
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4002863