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Journal of Operator Theory

Volume 62, Issue 2, Fall 2009  pp. 281-295.

Composition operators from weak to strong spaces of vector-valued analytic functions

Authors Jussi Laitila (1), Hans-Olav Tylli (2), and Maofa Wang (3)
Author institution: (1) Department of Mathematics and Statistics, University of Helsinki, P.B. 68 (Gustaf H\"allstr\"omin katu 2b), FIN-00014 University of Helsinki, Finland
(2) Department of Mathematics and Statistics, University of Helsinki, P.B. 68 (Gustaf H\"allstr\"omin katu 2b), FIN-00014 University of Helsinki, Finland
(3) School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China


Summary:  Let φ be an analytic map from the unit disk into itself, X a complex infinite-dimensional Banach space and 2. It is shown that the composition operator C_\varphi\colon f \mapsto f \circ \varphi is bounded wH^p(X) \to H^p(X) if and only if C_\varphi is a Hilbert--Schmidt operator H^2 \to H^2. Here H^p(X) is the X-valued Hardy space and wH^p(X) is a related weak vector-valued Hardy space. A similar result is established for vector-valued Bergman spaces.


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