*(English)*Zbl 1119.47022

Let $\phi $ be an analytic map of the unit disk $\mathbb{D}$ into itself, and denote by ${h}^{1}$, ${H}^{1}$, $P{L}^{1}$, and $CT$ the harmonic Hardy space of the unit circle $T$, the analytic Hardy space, the space of Poisson integrals of ${L}^{1}(\mathbb{T},dm)$ functions, and the space of Cauchy transforms of measures on the unit circle, respectively. The composition operator ${C}_{\phi}$, defined by ${C}_{\phi}f=f\circ \phi $, is bounded on each of these spaces.

The following well-known theorem was obtained by combining the efforts of D. Sarason with those of J. H. Shapiro and C. Sundberg. The operator ${C}_{\phi}$ is compact on ${h}^{1}$ if and only if it is compact on $P{L}^{1}$, if and only if it is such on ${H}^{1}$; any of these conditions is further equivalent to

as well as to the weak compactness on any of the spaces mentioned. Here, $\phi $ actually denotes the radial limit function of $\phi $, defined almost everywhere on $\mathbb{T}$. (Related partial results also exist for the space $CT$ and are due to P. Bourdon, J. Cima, and A. Matheson.)

Inspired by the growing interest in composition operators in a vector-valued setting, as a main result, the authors of the paper under review extend the above characterization to a more general context. Namely, it is possible to define ${C}_{\phi}$ on a complex Banach space $X$ and the functions $f:\mathbb{D}\to X$, as well as the vector-valued harmonic Hardy space ${h}^{1}\left(X\right)$ and the space $CT\left(X\right)$ of vector-valued Cauchy transforms.

The authors prove in this nice paper that ${C}_{\phi}$ is weakly compact on either space ${h}^{1}\left(X\right)$, $CT\left(X\right)$ if and only if the above condition ($*$) holds and $X$ is reflexive. (It should be remarked that the operator cannot be compact as long as $X$ is infinite-dimensional.) The difficult part is to prove the statement under the assumption that $X$ is reflexive. The authors employ a clever argument which allows them to avoid discussing the existence of radial limits of the functions in ${h}^{p}\left(X\right)$, among other numerous technical tools, many of them of abstract functional-analytic nature.

The paper also extends and complement recent results by Bonet, Domański, and Lindström and by Blasco for composition operators on weak spaces $w{H}^{p}\left(X\right)$ and $w{h}^{p}\left(X\right)$ of vector-valued harmonic functions. Examples are also provided that show that for any complex infinite-dimensional Banach space $X$ and $1\le p<\infty $, there exist functions which are in $w{H}^{p}\left(X\right)$ but not in ${H}^{p}\left(X\right)$ (resp., in $w{h}^{p}\left(X\right)$ but not in $w{h}^{p}\left(X\right)$). The paper also contains several other interesting results, examples and comments.