  Suppose $(C_t)_{t\geq0}$ is the composition semigroup induced by a
one-parameter semigroup $(\varphi_t)_{t\geq0}$ of analytic self-maps of the
unit disk. The main purpose of the paper is to investigate the spectrum of the
infinitesimal generator of $(C_t)_{t\geq0}$ acting on the weighted Bergman
space induced by doubling weights, provided $(\varphi_t)_{t\geq0}$ 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
$(C_t)_{t\geq0}$ 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
$(C_t)_{t\geq0}$.
