TAILIEUCHUNG - Báo cáo toán học: "On the structure of the Naimark dilation "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên cấu trúc của sự giãn nở Naimark. | J. OPERATOR THEORY 12 1984 159-175 Copyright by INCREST 1984 ON THE STRUCTURE OF THE NAIMARK DILATION T. CONSTANTINESCU One of the main ideas of classical work of I. Schur 9 is to give a concrete description of the structure of Fourier coefficients of a scalar contractive analytic function on the unit circle by means of an associated sequence of complex numbers Tils Ll with the properties that y 1 for every n 6 N and if y I 1 then y 0 for n nữ . For the operator-valued case of this problem one needs a good generalization of such kind of sequence this was done in 4 by introducing the notion of choice sequences see Section 1 for the definition . There is a large class of problems intimately connected with the previous Schur problem. Let us mention some extrapolation problems the contractive intertwining dilations theory the structure of positive Toeplitz forms the Naimark dilation of semispectral measures and so on. Choice sequences appear in all of these and seem to be a powerful tool in dealing with some specific problems of these topics. The aim of this paper is to present the structure of the connection between the Naimark dilation of operator-valued semispectral measures on the unit circle and choice sequences. We use for this some improvements of the techniques developed in our previous paper 7 on the structure of positive Toeplitz forms. The matrix form of the Naimark dilation can be thought as a generalization of the Schaffer form of s unitary dilation of a contraction 11 . We must also mention that w in is in fact what is called in 5 an adequate isometry. As an application we compute the prediction-error operator of a stationary process. 1 We shall begin this section by reminding a series of facts and notation from 7 and some usual notation in the contractive intertwining dilations theory. Thus let 3 and be two Hilbert spaces and let Te Jf be a contraction . 1 7 11 SỈ 1 . We denote as usual DT I r T V2 and S T DT3 the defect operator and

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