Contractions and the Commutant Lifting Theorem in Kreĭn Spaces

A brief survey of the commutant lifting theorem is presented. This is initially done in the Hilbert space context in which the commutant lifting problem was initially considered, both in Sarason’s original form and that of the later generalization due to Sz.-Nagy and Foias. A discussion then follows of the connection with contraction operator matrix completion problems, as well as with the Sz.-Nagy and Andô dilation theorems. Recent work in abstract dilation theory is outlined, and the application of this to various generalizations of the commutant lifting theorem are indicated. There is a short survey of the relevant Kreĭn space operator theory, focusing in particular on contraction operators and highlighting the fundamental differences between such operators on Kreĭn spaces and Hilbert spaces. The commutant lifting theorem is formulated in the Kreĭn space context, and two proofs are sketched, the first using a multistep extension procedure with a Kreĭn space version of the contraction operator matrix completion theorem, and the 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.