Composition operators on weighted Hardy spaces

Abstract

Suppose $\varphi$ is an analytic self-map of open unit disk $\mathbf{D}$ and $\psi$ is an analytic function on $\mathbf{D}$. Then a weighted composition operator induced by $\varphi$ with weight $\psi$ is given by $(W_{\psi, \varphi}f)(z)= \psi(z)f(\varphi(z))$, for $z \in \mathbf{D}$ and $f$ analytic on $\mathbf{D}$. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators $W_{\psi, \varphi}$. In terms of fixed points in the closed unit disk $\overline{\mathbf{D}}$, conditions under which $W_{\psi, \varphi}$ is compact are given. Necessary conditions for the compactness of $C_{\varphi}$ are given in terms of the angular derivative $\varphi^{\prime}(\zeta)$ where $\zeta$ is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten $p$-class $S_{p}(H^s(\beta_1), H^q(\beta_2))$, where the inducing map has supremum norm strictly smaller than~$1$.