# 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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