Coupling and relaxed commutant lifting

TL;DR

This paper provides a comprehensive description of all contractive solutions to the relaxed commutant lifting problem using Schur class functions, extending classical results and identifying conditions for uniqueness.

Contribution

It introduces a Redheffer type description combining isometric coupling methods with realization results for the relaxed commutant lifting problem.

Findings

Provides a parameterization of solutions in special cases.

Shows non-uniqueness of solutions in general.

Identifies conditions for unique solutions.

Abstract

A Redheffer type description of the set of all contractive solutions to the relaxed commutant lifting problem is given. The description involves a set of Schur class functions which is obtained by combining the method of isometric coupling with results on isometric realizations. For a number of special cases, including the case of the classical commutant lifting theorem, the description yields a proper parameterization of the set of all contractive solutions, but examples show that, in general, the Schur class function determining the contractive lifting does not have to be unique. Also some sufficient conditions are given guaranteeing that the corresponding relaxed commutant lifting problem has only one solution.