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Redheffer Representations and Relaxed Commutant Lifting

Research paper by S. ter Horst

Indexed on: 09 Feb '10Published on: 09 Feb '10Published in: Complex Analysis and Operator Theory



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 \({\mathbb {D}}\) 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 the Redheffer representation level.