Let be an analytic map of the unit disk into itself, and denote by , , , and the harmonic Hardy space of the unit circle , the analytic Hardy space, the space of Poisson integrals of functions, and the space of Cauchy transforms of measures on the unit circle, respectively. The composition operator , defined by , 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 is compact on if and only if it is compact on , if and only if it is such on ; any of these conditions is further equivalent to
as well as to the weak compactness on any of the spaces mentioned. Here, actually denotes the radial limit function of , defined almost everywhere on . (Related partial results also exist for the space 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 on a complex Banach space and the functions , as well as the vector-valued harmonic Hardy space and the space of vector-valued Cauchy transforms.
The authors prove in this nice paper that is weakly compact on either space , if and only if the above condition () holds and is reflexive. (It should be remarked that the operator cannot be compact as long as is infinite-dimensional.) The difficult part is to prove the statement under the assumption that is reflexive. The authors employ a clever argument which allows them to avoid discussing the existence of radial limits of the functions in , 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 and of vector-valued harmonic functions. Examples are also provided that show that for any complex infinite-dimensional Banach space and , there exist functions which are in but not in (resp., in but not in ). The paper also contains several other interesting results, examples and comments.