Isometric Dilations for Infinite Sequences of Noncommuting Operators

Abstract

Abstract.This paper develops a dilation theory for {Tn}^Lx an infinite sequence of noncommuting operators on a Hubert space, when the matrix [Tx, Tj, ...] is a contraction. A Wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for {Tn}^Lx 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-Foias. lifting theorem to this noncommutative setting.This paper is a continuation of [5] and develops a dilation theory for an infinite sequence {Tn}f=x of noncommuting operators on a Hubert space ß?when E~ i T"T*n ^ Isc (fV is the identity on %*) ■ Many of the results and techniques in dilation theory for one operator [8] and also for two operators [3,4] are extended to this setting.First we extend Wold decomposition [8,4] to the case of an infinite sequence {Vn}°f=x of isometries with orthogonal final spaces.In §2 we obtain a minimal isometric dilation for {Tn}°f=x by extending the Schaffer construction in [6,4]. Using these results we give some theorems on the geometric structure of the space of the minimal isometric dilation. Finally, we give some sufficient conditions on a sequence {Tn}°f=x to be simultaneously quasi-similar to a sequence {Rn}f=x of isometries on a Hubert space S¡A with E~i*X = fV In §3 we use the above-mentioned theorems to obtain the Sz.-Nagy-Foias. lifting theorem [7,8,1,4] in our setting.In a subsequent paper we will use the results of this paper for studying the "characteristic function" associated to a sequence {Tn}fLx with Yff=\ T"T* < far • 1 Throughout this paper A stands for the set {1,2, ... ,k} (k e N) or the set N = {1,2,...}.

Structure and Entropy for Positive-Definite Toeplitz Kernels on Free Semigroups

“…be the wandering subspace in the Wold type decomposition for fV 1 ; : : : ; V n g (see Po1]). Denote G := 2F + n V L, V gi := V i ; i = 1; : : : ; n, and de ne the unitary operator U L : G !…”

Standard commuting dilations and liftings

“…We remark that Popescu's presentation (Equation (2.1) of [Po89]) for the mid was employed in [BBD04] for proving the above result. From the easy observation that the maximal commuting subspace of a row contraction T is a V -coinvariant subspace we conclude that V to be an isometric lifting of T • .…”

“…Every row contraction admits a minimal isometric dilation (mid for short) (cf. [Po89]) and it is unique up to unitary equivalence. An extensive theory is given by Popescu (e.g.…”

“…An extensive theory is given by Popescu (e.g. [Po89], [Po99]) for row contractions, that extends part of the harmonic analysis of contractions developed by Sz.Nagy and Foias (cf. [NF70]).…”

Complete Pick positivity and unitary invariance

“…For a contractive tuple or a row contraction, commuting or not, dilation theory has been worked out in great detail in the last two decades (see Arveson [6], Popescu [14] and [15], Müller and Vasilescu [13], and [8] and [9]). For other domains, Ambrozie, Engliš and Müller in [4] and Arazy and Engliš in [5] construct dilations under several conditions on the kernel function.…”