Hilbert-Schmidt differences of composition operators on the Bergman space

In the setting of the weighted Bergman space over the unit disk, we characterize Hilbert-Schmidt differences of two composition operators in terms of integrability condition involving pseudohyperbolic distance between the inducing functions. We also show that a linear combination of two composition operators can be Hilbert-Schmidt, except for trivial cases, only when it is essentially a difference. We apply our results to study the topological structure of the space of all composition operators under the Hilbert-Schmidt norm topology. We first characterize components and then provide some sufficient conditions for isolation or for non-isolation.