Relaxation of metric constrained interpolation and a new lifting theorem
This long, very interesting paper is a natural continuation of the masterpiece [``Metric constrained interpolation, commutant lifting and systems'' (Oper. Theory, Adv. Appl. 100, Birkhäuser, Basel) (1998; Zbl 0923.47009)] that was written by \textit{I.~Gohberg} together with the authors of the present paper. In the present paper, the authors give an explicit solution to a new lifting interpolation problem that includes the commutant lifting theorem and some of its generalizations, such as the weighted commutant lifting of \textit{A. Biswas, C. Foias} and \textit{A. E. Frazho} [Acta Sci. Math. 65, 657--686 (1999; Zbl 0954.47018)] and the fixed point approach to Nehari's problem of \textit{S. Treil} and \textit{A. Volberg} [Oper. Theory, Adv. Appl. 71, 165--186 (1994; Zbl 0818.47026)]. The importance of this main result is shown by way of a number of applications to different variants of interpolation problems of Schur--Carathéodory, Nevanlinna--Pick, and Sarason's types, among others. Precise statements are too involved to be described in a short review; we simply recommend the reading of this paper to any operator theorist.
scientific article; zbMATH DE number 1760618