Compact composition operators on Hardy-Orlicz spaces

Abstract

In this paper, compact composition operators acting on Hardy-Orlicz spaces $$H^{\Phi} = \big\{\, f ın H({\Bbb D}) : \sup_{0 < r < 1} ınt_{\partial {\Bbb D}} \Phi(\log^{+} |f(r e^{i \theta})|)\, d \sigma < ınfty \,\big\} $$ are studied. In fact, we prove that if $\Phi$ is a twice differentiable, non-constant, non-decreasing non-negative, convex function on $\Bbb R$, then the composition operator $C_{\varphi}$ induced by a holomorphic self-map $\varphi$ of the unit disk is compact on Hardy-Orlicz spaces $H^{\Phi}$ if and only if it is compact on the Hardy space $H^{2}$.