Hilbert-Schmidtness of weighted composition operators and their differences on Hardy spaces

Abstract

Let $u$ and $\varphi$ be two analytic functions on the unit disk $\mathbb{D}$ such that $\varphi\mathbb{D} \subset \mathbb{D}$. A weighted composition operator $uC_{\varphi}$ induced by $u$ and $\varphi$ is defined on $H^2$, the Hardy space of $\mathbb{D}$, by $uC_{\varphi}f := u \cdot f \circ \varphi$ for every $f$ in $H^2$. We obtain sufficient conditions for Hilbert-Schmidtness of $uC_{\varphi}$ on $H^2$ in terms of function-theoretic properties of $u$ and $\varphi$. Moreover, we characterize Hilbert-Schmidt difference of two weighted composition operators on $H^2$.