Contractions and the Commutant Lifting Theorem in Kreı̆n Spaces

A brief survey of the commutant lifting theorem is presented. This is initially done in the Hilbert spaces, the context in which the commutant lifting problem was first considered. Both Sarason’s original form and the later generalization of Sz.-Nagy and Foias are given. A discussion then follows of the connection with contraction operator matrix completion problems, as well as with the Sz.-Nagy and Andô dilation theorems. Contemporary work in abstract dilation theory is outlined, and the application of this to various generalizations of the commutant lifting theorem is indicated. There is a short survey of the relevant Kreı̆n space operator theory, focusing in particular on contraction operators and highlighting the fundamental differences between such operators on Kreı̆n spaces and Hilbert spaces. The commutant lifting theorem is then formulated in the Kreı̆n space context, and two proofs are sketched, the first using a multistep extension procedure with a Kreı̆n space version of the contraction operator matrix completion theorem, and a second diagrammatic approach which is a variation on a method due to Arocena. Finally, the problem of lifting intertwining operators which are not necessarily contractive is mentioned, as well as some open problems.