2019/12/07 by Daniel Alpay, Alpay, Daniel, Ariel Pinhas +3
Mathematics · #30F15 #46E22 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1912.03542
openalex publication_date 2019/12/07 · openalex created_date 2019/12/13 · openalex updated_date 2026/07/28
Carathéodory functions, i.e. functions analytic in the open upper half-plane and with a positive real part there, play an important role in operator theory, 1D system theory and in the study of de Branges-Rovnyak spaces. The Herglotz integral representation theorem associates to each Carathéodory function a positive measure on the real line and hence allows to further examine these subjects. In this paper, we study these relations when the Riemann sphere is replaced by a real compact Riemann surface. The generalization of Herglotz's theorem to the compact real Riemann surface setting is presented. Furthermore, we study de Branges-Rovnyak spaces associated with functions with positive real-part defined on compact Riemann surfaces. Their elements are not anymore functions, but sections of a related line bundle.