Type: Article
Publication Date: 1990-09-01
Citations: 173
DOI: https://doi.org/10.2140/pjm.1990.145.117
Earl Berkson has shown that certain highly non-compact composition operators on the Hardy space H 2 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 hall 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.