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