Spectra of infinitesimal generators of composition semigroups on weighted Bergman spaces induced by doubling weights

Suppose (Cₜ)t≥0 is the composition semigroup induced by a one-parameter semigroup (φₜ)t≥0 of analytic self-maps of the unit disk. The main purpose of the paper is to investigate the spectrum of the infinitesimal generator of (Cₜ)t≥0 acting on the weighted Bergman space induced by doubling weights, provided (φₜ)t≥0 is elliptic. The method applied is a certain spectral mapping theorem and a characterization of the spectra of certain composition operators. Eventual norm-continuity of (Cₜ)t≥0 also plays an important role, which can be depicted in terms of studying the difference of two distinct composition operators. As a byproduct, we also characterize a certain compact integral operator that is closely related to the resolvent of the infinitesimal generator of (Cₜ)t≥0.