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 ↦ C ϕ ( f ) = f ∘ ϕ exhibits a rather strong rigidity of non-compact behaviour on the Hardy space H p , for 1 ≤ p < ∞ and p ≠ 2 . Our main result is the following trichotomy, which states that exactly one of the following alternatives holds: (i) C ϕ is a compact operator H p → H p , (ii) C ϕ fixes a (linearly isomorphic) copy of ℓ p in H p , but C ϕ does not fix any copies of ℓ 2 in H p , (iii) C ϕ fixes a copy of ℓ 2 in H p . Moreover, in case (iii) the operator C ϕ actually fixes a copy of L p ( 0 , 1 ) in H p provided p > 1 . We reinterpret these results in terms of norm-closed ideals of the bounded linear operators on H p , which contain the compact operators K ( H p ) . In particular, the class of composition operators on H p does not reflect the quite complicated lattice structure of such ideals. [ABSTRACT FROM AUTHOR]