Isometric Dilations for Infinite Sequences of Noncommuting Operators

Abstract

This paper develops a dilation theory for $\{ {T_n}\} _{n = 1}^\infty$ an infinite sequence of noncommuting operators on a Hilbert space, when the matrix $[{T_1},{T_2}, \ldots ]$ is a contraction. A Wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for $\{ {T_n}\} _{n = 1}^\infty$ 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.