Weighted composition operators between n-th α-weighted spaces

Abstract

Let $V_n^{\alpha }$ be the Banach space of holomorphic functions on the open unit disk $\mathbb{𝔻}$ in the complex plane consisting of those $f$ such that and $V_{n,0}^{\alpha }$ be the closed subspace of $V_n^{\alpha }$ consisting of those $f$ for which ${lim}_{\left|z\right|\to 1}{\left(1-|z{|}^2\right)}^{\alpha }\left|f^{\left(n\right)}\left(z\right)\right|=0$, where $n$ is any nonnegative integer and $\alpha >0$. We give boundedness characterizations, norm estimates and essential norm estimates of weighted composition operators $W_{\psi ,\phi }:V_n^{\alpha }\to V_m^{\beta }$ and $W_{\psi ,\phi }:V_{n,0}^{\alpha }\to V_{m,0}^{\beta }$, respectively, where $W_{\psi ,\phi }f\left(z\right)=\psi \left(z\right)f\left(\phi \left(z\right)\right)$. As a corollary, we characterize the compactness of $W_{\psi ,\phi }$. Specifically, our characterizations involve not only the classical Julia–Carathéodory type condition, but also the powers ${\phi }^k$. In addition, our results extend several well-known results in the literature.