Composition operators on vector-valued analytic function spaces: a survey

Abstract

Abstract. We survey recent results about composition operators induced by analytic self-maps of the unit disk in the complex plane on various Banach spaces of analytic functions taking values in infinitedimensional Banach spaces. We mostly concentrate on the research line into qualitative properties such as weak compactness, initiated by Liu, Saksman and Tylli (1998), and continued in several other papers. We discuss composition operators on strong, respectively weak, spaces of vector-valued analytic functions, as well as between weak and strong spaces. As concrete examples, we review more carefully and present some new observations in the cases of vector-valued Hardy and BM OA spaces, though the study of composition operators has been extended to a wide range of spaces of vector-valued analytic functions, including spaces defined on other domains. Several open problems are stated.

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

“…See [2,16] for the above results. The first aim of this paper is to extend such general necessary and sufficient conditions to weighted composition operators between two different function spaces, where the weight function ψ is scalar valued, i.e., ψ ∈ H(D, C).…”

“…Assume that the following properties hold for the pair (A, A(X)) for all Banach spaces X (see [16,Section 2]):…”

“…Propositions 1 and 2 apply to a large number of function spaces, such as X-valued Hardy, Bergman, BMOA, and Bloch spaces as well as to their weak versions (see, for example, [16] for the definitions of these function spaces). Before explicitly discussing any corollaries, let us consider the general class of weak vector-valued function spaces (1.2), introduced by Bonet, Domański and Lindström [2] in the context of composition operators.…”

“…Both of these spaces satisfy (a2) and (b2), so the associated weak spaces wH 1 (X) and wA 1 (X) are Banach spaces for all X. However, they are quite different from the usual "strong" spaces H 1 (X) and A 1 (X); see [12,13,15].…”

“…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].…”