Bi-isometries and commutant lifting

Abstract

In a previous paper, the authors obtained a model for a bi-isometry, that is, a pair of commuting isometries on complex Hilbert space. This representation is based on the canonical model of Sz.-Nagy and the third author. One approach to describing the invariant subspaces for such a bi-isometry using this model is to consider isometric intertwining maps from another such model to the given one. Representing such maps requires a careful study of the commutant lifting theorem and its refinements. Various conditions relating to the existence of isometric liftings are obtained in this note, along with some examples demonstrating the limitations of our results.