On dilation and commuting liftings of ${\displaystyle n}$-tuples of commuting Hilbert space contractions

The ${\displaystyle n}$-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 ${\displaystyle n}$-tuple. A series of such liftings leads to an isometric dilation of the ${\displaystyle n}$-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 ${\displaystyle n}$-tuple has an isometric dilation.