Composition operators isolated in the uniform operator topology

Abstract:

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