Key points are not available for this paper at this time.
Suppose (Cₜ) ₓ₀ is the composition semigroup induced by a one-parameter semigroup (ₜ) ₓ₀ 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) ₓ₀ acting on the weighted Bergman space induced by doubling weights, provided (ₜ) ₓ₀ 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) ₓ₀ 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) ₓ₀.
Qian et al. (Wed,) studied this question.