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

Daniel Alpay; Ariel Pinhas; Victor Vinnikov

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arXiv · 1912.03542

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

Daniel Alpay·Ariel Pinhas·Victor Vinnikov

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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.

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Daniel Alpay, Ariel Pinhas, Victor Vinnikov. 2019-12-07. Carath\'eodory functions on Riemann surfaces and reproducing kernel spaces. https://arxiv.org/abs/1912.03542

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