Abstract
A class of finite dimensional reproducing kernel Krein spaces of vector valued rational functions M X with an indefinite inner product that is defined in terms of a singular Hermitian matrix X is analyzed. It is shown that if X is positive semidefinite, then M X is a reproducing kernel Hilbert space of the kind that originates in the work of L. de Branges if and only if X is a solution of an associated Riccati equation and a certain invariance condition (which is automatically fulfilled in some cases) is met. Analogous conclusions are obtained for the case when X is Hermitian and M X is a Krein space.
To Israel Gohberg: teacher, colleague and valued friend, with admiration and affection, on his seventieth
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Dym, H. (2001). On Riccati Equations and Reproducing Kernel Spaces. In: Dijksma, A., Kaashoek, M.A., Ran, A.C.M. (eds) Recent Advances in Operator Theory. Operator Theory: Advances and Applications, vol 124. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8323-8_10
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DOI: https://doi.org/10.1007/978-3-0348-8323-8_10
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