Composition operators on vector-valued harmonic functions and Cauchy transforms
Indiana University Mathematics Journal
Let ϕ be an analytic self-map of the unit disk. The weak compactness of the composition operators Cϕ : f → f •ϕ 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 Cϕ on certain spaces wh 1 (X) and wCT (X)
... h 1 (X) and wCT (X) of weak type by extending an alternative approach due to Bonet, Domański and Lindström. Concrete examples based on minimal prerequisites highlight the differences between h p (X) (respectively, CT (X)) and the corresponding weak spaces. [CM1, p. 61] for a convenient description. Bourdon and Cima [BC] observed that every composition operator C ϕ is bounded on CT , the Banach space consisting of the Cauchy transforms of Borel measures on the unit circle T. Later Cima and Matheson [CM2] proved that if C ϕ is weakly compact on CT then it is compact, and that the compactness of C ϕ on CT is equivalent to its compactness on H 1 . Currently there is growing interest in the composition operators C ϕ in a vector-valued setting. For instance, the boundedness and the weak compactness of f → f • ϕ have been studied on various spaces consisting of vectorvalued analytic functions f : D → X, such as the Hardy spaces H p (X) [LST], the (weighted) Bergman and Bloch spaces [LST], [BDL], and BM OA(X) [L]. Here X is a complex infinite-dimensional Banach space. The reference [BF] discusses a weighted locally convex setting, and other results related to vector-valued Hardy spaces are found in [HJ] and [SB]. The aim of this paper is to add to the picture from [LST], [BDL] and [L] by studying composition operators on the harmonic Hardy spaces h p (X) and the space CT (X) of vector-valued Cauchy transforms. We establish a version of the Omnibus Theorem for h 1 (X), and of the results of [CM2] for CT (X), by showing that C ϕ is weakly compact on h 1 (X) (respectively, on CT (X)) if and only if condition (5) holds and X is reflexive. (Note that C ϕ is never compact on these vector-valued spaces once X is infinite-dimensional, since C ϕ preserves the constant functions f x (z) ≡ x for x ∈ X.) A novel feature in the vector-valued setting is that there usually are more than one canonical X-valued space which correspond to a given classical scalar-valued space. In section 5 we characterize the weakly compact composition operators on a class wE(X) of weak spaces of vector-valued harmonic functions. This complements analogous results by Bonet, Domański and Lindström [BDL] for weak spaces of analytic functions. Our result applies for instance to the weak harmonic Hardy space wh 1 (X) considered by e.g.