Earl Berkson has shown that
certain highly non-compact composition operators on the Hardy space H2
are, in the operator norm topology, isolated from all the other composition
operators. On the other hand, it is easy to see that no compact composition
operator is so isolated. Here we explore the intermediate territory, with the
following results: (i) Only the extreme points of the H∞ unit ball can induce
isolated composition operators. In particular, those holomorphic self-maps of
the unit disc whose images make at most finite order of contact with the
unit circle induce composition operators that are not isolated. However, (ii)
extreme points do not tell the whole story about isolation: some of them induce
compact, hence non-isolated, composition operators. Nevertheless, (iii) all
sufficiently regular univalent extreme points induce isolated composition
operators.
