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

$\varphi$ be an analytic map from the unit disk into itself,

$X$ a complex infinite-dimensional Banach space and

$2 \leqslant p < \infty$ . 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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