Weighted Composition Operators on Hardy Spaces

Abstract

Ž .Let , be analytic functions defined on ‫,ބ‬ such that ‫ބ‬ : ‫.ބ‬ The operator Ž . given by f ¬ f ( is called a weighted composition operator. In this paper we deal with the boundedness, compactness, weak compactness, and complete continu-Ž . ity of weighted composition operators on Hardy spaces H 1 F p -ϱ . In p particular, we prove that such an operator is compact on H if and only if it is 1 weakly compact on this space. This result depends on a technique which passes the weak compactness from an operator T to operators dominated in norm by T.

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

“…In [7] it was further shown that every weakly compact W ψ,ϕ on H 1 is compact. Moreover, because A 1 is isomorphic to ℓ 1 , it has the Schur property, and so every weakly compact linear operator on A 1 is compact; see [19, p. 302].…”

“…Compactness and weak compactness of W ψ,ϕ are well understood on H 1 and A 1 : Function-theoretic characterizations exist for the compactness of W ψ,ϕ on both H 1 and A 1 [7,9]. In [7] it was further shown that every weakly compact W ψ,ϕ on H 1 is compact.…”

Norm Estimates for Weighted Composition Operators on Spaces of Holomorphic Functions

“…However, this is not a necessary and sufficient condition for boundedness, although clearly we require h ∈ H 2 (D) (see [9] for more details). Weighted composition operators on Hardy and Bergman spaces have been much studied in recent years (see, for example, the articles [1,2,3,4,6,9,11] and the survey [16]). In particular, their boundedness on H 2 (D) is characterised in [3,6,11], as is explained further in Section 2.…”

Weighted composition operators between spaces of Dirichlet type

“…We also obtain estimates for the essential norm of weighted composition operators on these spaces. For weighted composition operators on spaces of holomorphic functions one can refer to [2,4,11,17,20,23] and the references therein.…”