Operator-weighted composition operators between weighted spaces of vector-valued analytic functions

Abstract

Abstract. We investigate several properties of operator-weighted composition mapsspaces of vector-valued analytic functions on the unit disc D. Here ϕ is an analytic self-map of D and ψ is an analytic operator-valued function on D. We characterize when the operator is continuous, maps a neighbourhood into a bounded set or maps bounded sets into relatively compact sets. In this way we extend results due to Laitila and Tylli for the case of Banach valued functions. This more general setting permits us to compare the results in the unweighted and weighted cases. New examples are provided, especially when the spaces X and Y are Köthe echelon spaces. They show the differences between the present setting and the case of functions taking values in Banach spaces.

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

“…The present knowledge of operator-weighted compositions is fairly rudimentary and most of the results deal with weighted H ∞ v (X) spaces and their locally convex variants. Characterizations of boundedness and (weak) compactness of W ψ,ϕ between H ∞ v (X) spaces were obtained in [29], and results for certain locally convex spaces of analytic vector-valued functions are found in [37] and [6]. Boundedness of the operator multipliers M ψ in related settings has been considered e.g.…”

“…Here H 0 w (L(X, Y ))) denotes the closure of the analytic L(X, Y )-valued polynomials in H ∞ w (L(X, Y ))), and W (X, Y ) the weakly compact operators X → Y . There are further differences between H ∞ v (X) and the locally convex setting as studied in [6].…”

Weighted Vector-Valued Holomorphic Functions on Banach Spaces

“…Proof. First, we proceed similarly as in [30,Lemma 21] to get a sequence ( ) in V 0 such that ( ) converges to 0 in , and there exists > 0 such that ‖ ‖ V ≥ for all ∈ N.…”

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

“…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]. The study of composition operators on vector-valued function spaces involves some basic results which hold for large classes of function spaces.…”