Alpay, Daniel; Bruinsma, Piet; Dijksma, Aad; de Snoo, Henk Interpolation problems, extensions of symmetric operators and reproducing kernel spaces. I. (English) Zbl 0737.47016 Topics in matrix and operator theory, Proc. Workshop, Rotterdam/Neth. 1989, Oper. Theory, Adv. Appl. 50, 35-82 (1991). [For the entire collection see Zbl 0722.00022.]The aim of the paper is to study interpolation problems for pairs of functions of the extended Nevanlinna class using two different approaches, namely the Krein-Langer theory of extensions of symmetric operators and the de Branges theory of Hilbert spaces of analytic functions, and to make explicit various links between them. (From the authors’ abstract.)In the first part, some properties of extended Nevanlinna classes in a Hilbert space are studied and the parametrizations of the solutions of the interpolation problem are proved using the Krein-Langer extension theory. Some results pertaining to the Lyapunov equation are presented. In the second part of the paper (to appear), those problems will be treated from the point of view of the de Branges theory. Reviewer: J.Durdil (Praha) Cited in 2 ReviewsCited in 17 Documents MSC: 47A57 Linear operator methods in interpolation, moment and extension problems 47A20 Dilations, extensions, compressions of linear operators 46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) 40A05 Convergence and divergence of series and sequences Keywords:Lyapunov equation; de Branges theory of Hilbert spaces of analytic functions; Krein-Langer theory of extensions of symmetric operators; interpolation problems for pairs of functions; extended Nevanlinna class Citations:Zbl 0722.00022 PDFBibTeX XMLCite \textit{D. Alpay} et al., in: Wiener-Hopf factorization in the inverse scattering theory for the \(n\)-D Schrödinger equation. . 35--82 (1991; Zbl 0737.47016)