Carath\'eodory functions on Riemann surfaces and reproducing kernel spaces

TL;DR

This paper extends the theory of Carathéodory functions and de Branges-Rovnyak spaces from the upper half-plane to compact real Riemann surfaces, generalizing classical integral representations and analyzing associated spaces.

Contribution

It generalizes Herglotz's theorem and studies de Branges-Rovnyak spaces on compact Riemann surfaces, introducing new geometric and functional analytic frameworks.

Findings

Generalization of Herglotz's theorem to Riemann surfaces

Characterization of de Branges-Rovnyak spaces on Riemann surfaces

Elements are sections of line bundles, not functions

Abstract

Carath\'eodory 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 Carath\'eodory 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.