Weighted composition operators between n-th α-weighted spaces

Abstract

Let Vnα be the Banach space of holomorphic functions on the open unit disk 𝔻 in the complex plane consisting of those f such that ∥ f ∥Vnα:=∑i=0n−1 f i0+supz∈D1−|z|2α fnz< ∞ and Vn,0α be the closed subspace of Vnα consisting of those f for which lim|z|→1(1 − |z|2)α |f (n)(z)|=0, where n is any nonnegative integer and α>0. We give boundedness characterizations, norm estimates and essential norm estimates of weighted composition operators Wψ,φ : Vnα→Vmβ and Wψ,φ : Vn,0α→Vm,0β, respectively, where Wψ,φ f (z)=ψ(z) f (φ(z)). As a corollary, we characterize the compactness of Wψ,φ. Specifically, our characterizations involve not only the classical Julia–Carathéodory type condition, but also the powers φk. In addition, our results extend several well-known results in the literature.