A General Completion Theorem

Abstract

In this chapter a general completion theorem, which may be viewed as a time-varying version of the commutant lifting theorem, is presented. Three different proofs are given. One proof uses the reduction techniques of Chapter X to convert the three chains completion theorem to a standard commutant lifting setup. The second proof goes by one step extensions, using Parrott’s lemma. The third proof gives an explicit formula for a solution which is the analogue of the central intertwining lifting. A nonstationary maximum entropy principle is given. Finally, as a first application, the three chains completion theorem is used to give a new proof of the Carswell-Schubert theorem.