TAILIEUCHUNG - Báo cáo toán học: "Isometric dilations of commuting contractions "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Lên dilations bóp đi lại. | J. OPERATOR THEORY 12 1984 . 65-88 Copyright by INCREST 1984 ISOMETRIC DILATIONS OF COMMUTING CONTRACTIONS. I ZOIA CEAUSESCU and I. SUCIU INTRODUCTION Let X be a Hilbert space and Ti iei be a system of commuting contractions Tị e X i 6 7 as usual for Tf Hilbert spaces L X X denotes the algebra of all linear bounded operators from 3 e to w and X denotes L X X . One calls an isometric dilation of the system Tjier any system Vi ieI of commuting isometries Vi iel acting on a Hilbert space X containing as a linear closed subspace such that T 1. T k . vnk I X 1 k 1 k I for nj 0 0 j k and for any finite subset of I P denoting for any subspace off of a Hilbert space X the orthogonal projection of X on X . If the system yjie consists of unitary operators then it is called a unitary dilation of Tjie . It is known that for any pair of commuting contractions on X there exists an isometric dilation see 1 and that for a system of commuting contractions consisting of more than two contractions an isometric dilation in general does not exist see 9 . This paper is concerned with the isometric dilations of a pair of commuting contractions. The first construction of such a dilation is due to T. Ando see 1 . Another known construction is based on the theorem of B. and c. Foias on the existence of a contractive dilation for any commutant of a given contraction see 11 . But none of these constructions can give all the isometric dilations of a pair of commuting contractions as some very simple examples can easily show it . The aim of this paper is to provide a construction of isometric dilations of a pair of commuting contractions which produces all the dilations Section 2 . Using this construction we establish in Section 3 a necessary and sufficient condition for uniqueness of the Ando dilation containing a given dilation of the commutant which permits to obtain directly the known result concerning the uniqueness of the Ando dilation of a pair of commuting contractions see 3 .

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