Small composition operators on analytic vector-valued function spaces

Abstract

Let φ be an analytic mapping of the unit disk D into itself. We characterize the weak compactness of the composition operator C φ : f → f • φ on the vector-valued Hardy space H 1 (X) (= H 1 (D, X)) and on the Bergman space B 1 (X), where X is a Banach space. Reflexivity of X is a necessary condition for the weak compactness of C φ in each case. Assuming this, the operator C φ :

Extension of vector-valued holomorphic and harmonic functions

“…Vector-valued holomorphic functions are useful in the theory of topological algebras [19] and of one-parameter semigroups [4,18], in infinite-dimensional holomorphy [15,Chapter 3], and in operator theory [1,20]. Composition operators on spaces of this type have been investigated recently [10,11,32,31]. The topic we consider is closely related to the investigation of conditions ensuring that a weakly holomorphic function with values in a locally convex space is holomorphic.…”

Composition operators on vector-valued harmonic functions and Cauchy transforms

“…Thus the space PL 1 a (X) is also implicit in the conditions below. The argument below also provides an alternative approach to [LST,Thm. 3.…”

“…One may also add the equivalent condition that C ϕ is weakly conditionally compact PL 1 a (X) → PL 1 a (X). We leave the details to the reader (see also [LST,Thm. 7]).…”

“…It follows that C ϕ is actually compact on h ∞ . Indeed, C ϕ is then weakly compact on the invariant subspace H ∞ , and hence compact on H ∞ (see the proof of [LST,Thm. 6]).…”

Weighted composition operators between weak spaces of vector-valued analytic functions

“…Weighted compositions appear naturally: for a large class of Banach spaces X, all linear onto isometries between X-valued H ∞ spaces are of the form (1.1) for suitable ψ and ϕ; see [4] and [18]. Properties of multipliers M ψ and composition operators C ϕ have been widely studied on various spaces of vector-valued analytic functions, including vector-valued Hardy, Bergman, BMOA and Bloch spaces; see, for example, [2,12,13,19,20] and the recent survey [16]. This study has further been extended to weighted composition operators; see [3,11,14,21,22,28].…”