A Generalized Schur–Takagi Interpolation Problem

Abstract

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.