A functional model associated with a generalized Nevanlinna pair

Abstract

Let \( \mathcal{L} \) be a Hilbert space, and let \( \mathcal{H} \) be a Pontryagin space. For every self-adjoint linear relation \( \tilde{A} \) in \( \mathcal{H} \oplus \mathcal{L} \), the pair \( \{ I + \lambda \psi (\lambda ),\,\psi (\lambda )\} \) where \( \psi (\lambda ) \) is the compressed resolvent of \( \tilde{A} \), is a normalized generalized Nevanlinna pair. Conversely, every normalized generalized Nevanlinna pair is shown to be associated with some self-adjoint linear relation \( \tilde{A} \) in the above sense. A functional model for this selfadjoint linear relation \( \tilde{A} \) is constructed.