×

Compact composition operators on Hardy-Orlicz spaces. (English) Zbl 1199.46073

For \(\phi:\mathbb R\to[0,\infty)\), which is twice differentiable, non-constant, non-decreasing and convex, let \(H^\phi\) be the Hardy-Orlicz space consisting of functions \(f\) holomorphic in the unit disc \(D\) for which \[ \| f\| _{H^\phi}=\sup_{0<r<1}\int_{\partial D}\phi(\log^+| f(re^{i\theta}| )\,d\theta<\infty. \] It is shown that the composition operator \(C_\varphi\) induced by a holomorphic self-map \(\varphi\) of the unit disc \(D\) is compact on \(H^\varphi\) if and only if it is compact on the Hardy space \(H^2\).

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions
47B33 Linear composition operators