Kernel-induced distance and its applications to Composition operators on Large Bergman spaces

In this paper, we obtain a complete characterization for the compact difference of two composition operators acting on Bergman spaces with a rapidly decreasing weight $\omega=e^{-\eta}$, $\Delta\eta>0$. In addition, we provide simple inducing maps which support our main result. We also study the topological path connected component of the space of all bounded composition operators on $A^2(\omega)$ endowed with the Hilbert-Schmidt norm topology.