Redheffer representations and relaxed commutant lifting

TL;DR

This paper explores the conditions under which Redheffer representations correspond to solutions of relaxed commutant lifting problems, providing necessary and sufficient criteria, and discusses non-uniqueness and harmonic maximal principles in this context.

Contribution

It establishes necessary and sufficient conditions for Redheffer representations to describe solutions of relaxed commutant lifting problems, and extends results on non-uniqueness and harmonic principles.

Findings

Characterization of Redheffer coefficients for commutant lifting solutions

Conditions for Redheffer representations to correspond to lifting problems

Generalization of harmonic maximal principle in this setting

Abstract

It is well known that the solutions of a (relaxed) commutant lifting problem can be described via a linear fractional representation of the Redheffer type. The coefficients of such Redheffer representations are analytic operator-valued functions defined on the unit disc D of the complex plane. In this paper we consider the converse question. Given a Redheffer representation, necessary and sufficient conditions on the coefficients are obtained guaranteeing the representation to appear in the description of the solutions to some relaxed commutant lifting problem. In addition, a result concerning a form of non-uniqueness appearing in the Redheffer representations under consideration and an harmonic maximal principle, generalizing a result of A. Biswas, are proved. The latter two results can be stated both on the relaxed commutant lifting as well as on on the Redheffer representation level.