Topological structure in the space of (weighted) composition operators on weighted Banach spaces of holomorphic functions

Abstract We consider the topological structure problem for the space of composition operators as well as the space of weighted composition operators on weighted Banach spaces with sup-norm. For the first space, we prove that the set of all composition operators that differ from the given one by a compact operator is path connected; however, in general, it is not always a component. Furthermore, we show that the set of compact weighted composition operators is path connected, but it is not a component in the second space.