Isometric dilations for infinite sequences of noncommuting operators

Abstract

This paper develops a dilation theory for { T n } n = 1 ∞ \{ {T_n}\} _{n = 1}^\infty an infinite sequence of noncommuting operators on a Hilbert space, when the matrix [ T 1 , T 2 , … ] [{T_1},{T_2}, \ldots ] is a contraction. A Wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for { T n } n = 1 ∞ \{ {T_n}\} _{n = 1}^\infty are obtained. Some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given. This results are used to extend the Sz.-Nagy-Foiaş lifting theorem to this noncommutative setting.