On dilation and commuting liftings of n -tuples of commuting Hilbert space contractions

The n -tuples of commuting Hilbert space contractions are considered. We give a model of a commuting lifting of one contraction and investigate conditions under which a commuting lifting theorem holds for an n -tuple. A series of such liftings leads to an isometric dilation of the n -tuple. The method is tested on some class of triples motivated by Parrotts example. It provides also a new proof of the fact that a positive definite n -tuple has an isometric dilation.