Topological Structure of the Spaces of Composition Operators on Hilbert Spaces of Dirichlet Series

In this paper we study some topological properties of the space of bounded composition operators on some Hilbert spaces of Dirichlet series. We first obtain formulas for the norms and essential norms of composition operators and differences of composition operators on Hilbert spaces of Dirichlet series. Then we give a characterization of the isolated points in the topological space of bounded composition operators on some Hilbert spaces of Dirichlet series. Finally we obtain sufficient conditions such that two composition operators are in the same path component. We show, among other results, that all compact composition operators are in the same path component. For a certain class of frequencies we give complete description of path components.