Compact composition operators on H2 and Hardy–Orlicz spaces

We compare the compactness of composition operators on H2 and on Orlicz–Hardy spaces HΨ. We show that, for every 1⩽p<∞, there exists an Orlicz function Ψ such that Hp+ε⊆HΨ⊆Hp for every ε>0, and a composition operator Cϕ which is compact on Hp and on Hp+ε, but not on HΨ. We also show that, for every Orlicz function Ψ which does not satisfy condition Δ2, there is a composition operator Cϕ which is compact on H2 but not on HΨ, and that, when Ψ grows fast enough, there is a function ϕ such that Cϕ is in all Schatten classes Sp, for p>0, but is not compact on HΨ.