Rigidity of composition operators on the Hardy space Hp

Let ϕ be an analytic map taking the unit disk D into itself. We establish that the class of composition operators f↦Cϕ(f)=f∘ϕ exhibits a rather strong rigidity of non-compact behaviour on the Hardy space Hp, for 1≤p<∞ and p≠2. Our main result is the following trichotomy, which states that exactly one of the following alternatives holds: (i) Cϕ is a compact operator Hp→Hp, (ii) Cϕ fixes a (linearly isomorphic) copy of ℓp in Hp, but Cϕ does not fix any copies of ℓ2 in Hp, (iii) Cϕ fixes a copy of ℓ2 in Hp. Moreover, in case (iii) the operator Cϕ actually fixes a copy of Lp(0,1) in Hp provided p>1. We reinterpret these results in terms of norm-closed ideals of the bounded linear operators on Hp, which contain the compact operators K(Hp). In particular, the class of composition operators on Hp does not reflect the quite complicated lattice structure of such ideals.