Isometric dilations for infinite sequences of noncommuting operators

Popescu, Gelu

doi:10.1090/S0002-9947-1989-0972704-3

Isometric dilations for infinite sequences of noncommuting operators
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Trans. Amer. Math. Soc. 316 (1989), 523-536 Request permission

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.

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Harmonic analysis on operators on Hilbert space