Hankel operators in the von Neumann-Schatten classes
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.
scientific article; zbMATH DE number 4002863