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,...}.

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,...}.

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