Inner functions and weighted composition operators on the Hardy-Hilbert space with the unbounded weights

Abstract

Let $\varphi$ be an analytic self-map of the open unit disk. It is given several sufficient conditions on $\varphi$ for which there is $u\in H^2 H^{\infty}$ such that the weighted composition operator $M_u C_\varphi$ on $H^2$ is bounded.