Components of linear fractional composition operators on spaces of holomorphic functions

Abstract

In this paper we consider the topological structure problem for the space C(X) of all composition operators on a space X of holomorphic functions over the unit disc, where X is one of the following function spaces: the Hardy space H2, Bergman spaces Aαp, the space H∞, weighted Banach spaces Hv with sup-norm, the classical Bloch space B, and weighted Bloch spaces Bv. In particular, we give a necessary and sufficient condition for two linear fractional composition operators to be in the same component of C(X). A characterization of isolated such composition operators in C(X) is also established.