Composition operators on weighted Bergman spaces induced by doubling weights

Given a doubling weight ω on the unit disk \(\mathbb{D}\) $$\mathbb{D}$$ , let A _ω ^p be the space of all the holomorphic functions f, where

\(\Vert f\Vert_{A_{\omega}^{p}}:=\left(\int_{\mathbb{D}}\vert f(z)\vert^{p}\ \omega(z)dA(z)\right)^{1/p}<\infty.\) $$\parallel f{\parallel }_{A_{}^p}:={({\int }_{\mathbb{D}}\mid f(z){\mid }^p(z)dA(z))}^{1/p}<\mathrm{\infty }.$$

We completely characterize the topological connectedness of the set of composition operators on A _ω ^p . As an application, we construct an interesting example which reveals that two composition operators on A _α ^p in the same path component may fail to have a compact difference and give a negative answer to the Shapiro-Sundberg question in the (standard) weighted Bergman space. In addition, we completely describe the central compactness of any finite linear combinations of composition operators on A _ω ^p in three terms: a Julia-Carathéodory-type function-theoretic characterization, a power-type characterization, and a Carleson-type measure-theoretic characterization.