2024/05/22 by Qian, Ruishen, Wu, Fanglei, Wulan, Hasi
#Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2405.13662
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.