A Generalized Schur–Takagi Interpolation Problem

• Published in: Integral equations and operator theory Vol. 80; no. 2; pp. 165 - 227
• Main Authors: Derkach, Vladimir, Dym, Harry
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01-10-2014
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Associated pair; Linear fractional transformation; Coprime factorization; Bitangential interpolation; Reproducing kernel space; de Branges spaces; Primary 30E05; Secondary 46E22; 47B50; Potapov-Ginzburg transform; 47A48; Kreĭn–Langer factorization; Resolvent matrix; Generalized Schur class
• ISSN: 0378-620X; 1420-8989
• DOI: 10.1007/s00020-014-2174-5

A generalized bitangential interpolation problem of Nevanlinna–Pick type in the class of generalized Schur matrix valued functions (mvf’s) is considered both in the unit disc and in the right half plane. Linear fractional representations of the set of solutions to these problems are presented in the strictly completely indeterminate case and in the singular case. These representations make use of a description of the ranges of linear fractional transformations with suitably chosen domains that was developed in Derkach and Dym (Integral Equations Operator Theory 65:1–50,
2009
). The set of solutions to the problem is parametrized by the set of Schur mvf’s, which satisfy some coprimeness conditions with respect to a given pair of inner mvf’s. Enroute, it is shown that the de Branges identity for the indefinite inner product based on a self-adjoint invertible operator
Q
holds if and only if
Q
is a solution of a Lyapunov–Stein equation. This extends an observation that was made earlier in a finite dimensional setting to Pontryagin spaces.