Rigidity of Composition Operators with Sum of Symbols on the Hardy Space H^(1+ϵ)

Abstract

Given (φ_1+φ_2) be sum of analytic maps taking the unit disk D into itself. We show follow [40] and establish that the class of composition operators f→ C_((φ_1+φ_2)) (f)= f∘(φ_1+φ_2) exhibits a rather strong rigidity of non-compact behaviour on the Hardy space H^(1+ϵ), for 0≤ϵ<∞ and ϵ≠1. The main result states that exactly one of the following alternatives holds: (i) C_((φ_1+φ_2)) is a compact operator H^(1+ϵ) → H^(1+ϵ), (ii) C_((φ_1+φ_2)) fixes a copy of 1+ϵ in H^(1+ϵ), but C_((φ_1+φ_2)) does not fix any copies of 2 in H^(1+ϵ), (iii) C_((φ_1+φ_2)) fixes a copy of 2 in H^(1+ϵ). In case (iii) the operator C_((φ_1+φ_2)) actually fixes a copy of L^(1+ϵ) (0,1) in H^(1+ϵ) provided ϵ>0. We reinterpret these results in terms of norm-closed ideals of the bounded linear operators on H^(1+ϵ), which contain the compact operators (1+ϵ)(H^(1+ϵ)). The class of composition operators on H^(1+ϵ) does not reflect the quite complicated lattice structure of