Topological structure of the space of composition operators on H∞ of Dirichlet series

In this paper, we study properties of the topological space of composition operators acting on the space of Dirichlet series. Especially, we show that there are two compact composition operators which are not in the same path component on H-infinity. This is in sharp contrast with the classical case where all compact composition operators H-infinity on of one variable or several variables lie in the same path component.