Abstract

In this paper, compact composition operators acting on Hardy-Orlicz spaces $$H^{\Phi} = \big\{\, f \in H({\Bbb D}) : \sup_{0 < r < 1} \int_{\partial {\Bbb D}} \Phi(\log^{+} |f(r e^{i \theta})|)\, d \sigma < \infty \,\big\} $$ 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 $H^{2}$.

Keywords: Hardy-Orlicz space, Composition operator, Nevanlinna counting function, vanishing Carleson measure.

MSC: 47B33, 46E38, 30D55

Pages:  215$-$224     

Volume  60 ,  Issue  3 ,  2008