Relaxed commutant lifting and a relaxed Nehari problem: Redheffer state space formulas

Abstract

The description of all solutions to the relaxed commutant lifting problem in terms of an underlying contraction, obtained earlier in joint work of the author with A.E. Frazho and M.A. Kaashoek, is transformed into a linear fractional Redheffer state space form. Under certain additional conditions the coefficient functions in this representation are described explicitly in terms of the original data. The main theorem is a generalization of the Redheffer description of all solutions to the classical commutant lifting problem. To illustrate the result a relaxed version of the Nehari extension problem is considered, and an explicit Redheffer description of all its solutions is given, assuming that a certain truncated Hankel operator is a strict contraction. The latter result is specified further for two special cases.