Linear connection between composition operators on the Hardy space

We consider the space of all composition operators, acting on the Hardy space over the unit disk, in the uniform operator topology. We obtain a characterization for linear connection between composition operators. As one of applications, we see that the set of all compact composition operators is a polygonally connected component, in sharp contrast to the known fact that this set is properly contained in a path connected component. When the inducing maps have “good” boundary behavior in the sense of higher-order data and order of contact, we extend/recover the Kriete-Moorhouse characterization for linear connection through a completely different approach relying on our results. We also notice some results in conjunction with the Bergman space case. Several questions motivated by our results are included.