Closed Range Weighted Composition Operators on $H^{2}$ and $A_{\alpha }^{2}$

Abstract

In this paper, first we investigate bounded below weighted composition operators $C_{\psi ,\varphi }$ on a Hilbert space of analytic functions. Then for $\psi \in H^{\infty }$ and a univalent map $\varphi$, we characterize all closed range weighted composition operators $C_{\psi ,\varphi }$ on $H^{2}$ and $A_{\alpha }^{2}$. Also we show that for $\psi \in H^{\infty }$ which is bounded away from zero near the unit circle, the weighted composition operator $C_{\psi ,\varphi }$ is bounded below on $H^{2}$ or $A_{\alpha }^{2}$ if and only if $C_{\varphi }$ has closed range. Moreover, we investigate invertible operators $C_{\psi _{1},\varphi _{1}}C_{\psi _{2},\varphi _{2}}^{*}$ and $C_{\psi _{1},\varphi _{1}}^{*}C_{\psi _{2},\varphi _{2}}$ on $H^{2}$ and $A_{\alpha }^{2}$.