Finite dimensional de Branges spaces on Riemann surfaces Academic Article uri icon

abstract

  • 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 α ; here we introduce generalized difference quotient operators R y α 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 α i and R y 2 α 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.

publication date

  • January 1, 2002