IUMJ

Title: Composition operators on vector-valued harmonic functions and Cauchy transforms

Authors: Jussi Laitila and Hans-Olav Tylli

Issue: Volume 55 (2006), Issue 2, 719-746

Abstract:

Let $\varphi$ be an analytic self-map of the unit disk. The weak compactness of the composition operators $C_{\varphi} \colon f \mapsto f \circ \varphi$ is characterized on the vector-valued harmonic Hardy spaces $h^1(X)$, and on the spaces $CT(X)$ of vector-valued Cauchy transforms, for reflexive Banach spaces $X$. This provides a vector-valued analogue of results for composition operators which are due to Sarason, Shapiro and Sundberg, as well as Cima and Matheson. We also consider the operators $C_{\varphi}$ on certain spaces $wh^1(X)$ and $w CT(X)$ of weak type by extending an alternative approach due to Bonet, Doma\'nski and Lindstr\"om. Concrete examples based on minimal prerequisites highlight the differences between $h^p(X)$ (respectively, $CT(X)$) and the corresponding weak spaces.