Lifting intertwining relations

Abstract

This note is devoted to the lifting problem for the intertwining relation $$XT_1^* = T_2 X,$$ where the T's are given contractions and X is a bounded operator. The lifting is possible if and only if X satisfies a stronger condition of boundedness introduced by the authors in a previous paper. The note includes a (partly expository) section which gives a simple proof of the commutant lifting theorem based on completing matrix contractions.