Journal of Operator Theory

Volume 55, Issue 2, Spring 2006 pp. 349-371.

Testing Schatten class Hankel operators, Carleson embeddings and weighted composition operators on reproducing kernels

Authors: Zen Harper (1) and Martin P. Smith (2)
Author institution: (1) Department of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
(2) Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom

Summary: Given an operator $A$ on a Hilbert space $\mathcal{H}$ and $c\in\mathcal{H}$, we consider operators $\Lambda_{A,c}$ defined on analytic functions $f$ by $\Lambda_{A,c} f = f(A)c$. Special cases of $\Lambda_{A,c}$ include vectorial Hankel operators, Carleson embeddings and weighted composition operators. For certain $A$, we determine conditions under which $\Lambda_{A,c}$ extends to an operator of Schatten von-Neumann class on the Hardy or Bergman space of the disc. These conditions involve only the action of $\Lambda_{A,c}$ on reproducing kernels and their derivatives. We also give corresponding results for operators on the Hardy space of the half-plane.

Keywords: chatten classes, reproducing kernels, Hardy space, Bergman space, Hankel operators, Carleson embeddings, composition operators.

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