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Finite Dimensional de Branges Spaces on Riemann Surfaces

✍ Scribed by Daniel Alpay; Victor Vinnikov

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
284 KB
Volume
189
Category
Article
ISSN
0022-1236

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✦ Synopsis

We study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real Riemann surface; these spaces are analogues of the spaces introduced by L. de Branges when the Riemann sphere is replaced by a compact real Riemann surface of a higher genus. In de Branges theory an important role is played by resolvent-like difference quotient operators R : ; here we introduce generalized difference quotient operators R y : for any non-constant meromorphic function y on the Riemann surface. The spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators R y 1 : i and R y 2 : 2 , where y 1 and y 2 generate the field of meromorphic functions on the Riemann surface, which satisfy a supplementary identity, analogous to the de Branges identity for difference quotients. Just as the classical de Branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint (or nonunitary) operator, the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks. 2002 Elsevier Science (USA)

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