Composition operators isolated in the uniform operator topology

Abstract

It is is shown that ϕ \phi is an analytic map of the disc | z | > 1 \left | z \right | > 1 into itself such that ϕ \phi has radial limits of modulus 1 on a set of positive measure, then for 1 ⩽ p > ∞ 1 \leqslant p > \infty the corresponding composition operator on H p {H^p} is isolated in the topological space of composition operators on H p {H^p} (with the uniform operator topology).