Isolation Amongst the Composition Operators

Joel H. Shapiro and Carl Sundberg

Annals of Mathematics 125 (1987), 375--404

Abstract: 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 composi-tion 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^infinity 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.

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