Compact composition operators on and Hardy–Orlicz spaces

We compare the compactness of composition operators on H 2 and on Orlicz–Hardy spaces H Ψ . We show that, for every 1 ⩽ p < ∞ , there exists an Orlicz function Ψ such that H p + ε ⊆ H Ψ ⊆ H p for every ε > 0 , and a composition operator C ϕ which is compact on H p and on H p + ε , but not on H Ψ . We also show that, for every Orlicz function Ψ which does not satisfy condition Δ 2 , there is a composition operator C ϕ which is compact on H 2 but not on H Ψ , and that, when Ψ grows fast enough, there is a function ϕ such that C ϕ is in all Schatten classes S p , for p > 0 , but is not compact on H Ψ .