Compact composition operators on $H^2$ and Hardy-Orlicz spaces

Abstract

We compare the compactness of composition operators on $H^2$ and on Orlicz-Hardy spaces $H^{\hat{I}}¨$. We show in particular that exists an Orlicz function $\hat{I}¨$ such that $H^{3+\eps} \subseteq H^Ψ\subseteq H^3$ for every $\eps >0$, and a composition operator $C_{\ddot{I}}•$ which is compact on $H^3$ and on $H^{3+\eps}$, but not compact on $H^{\hat{I}}¨$.

17 pages