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Spectral dilation of operator-valued measures and its application to infinite-dimensional harmonizable processes. (English) Zbl 0625.60042
The classical Naimark dilation theorem states that for any L(K,K)-valued semispectral measure T there exists a larger Hilbert space \(H\supset K\) and an L(H,H)-valued spectral measure E such that \(T(\cdot)=\Pr oj_ KE(\cdot)\), here \(\Pr oj_ K\) is the orthogonal projection from H to K and \(L(M,N)=\{A: M\to N|\) A bounded and linear\(\}\). Furthermore, according to a dilation theorem by the reviewer [Ann. Acad. Sci. Fenn., Ser. A I 3, 43-51 (1977; Zbl 0367.46041)] every Hilbert space - valued bounded vector measure can be represented as an orthogonal projection of a bounded orthogonally scattered vector measure with values in a larger Hilbert space.
The authors are concerned with the generalization of these results for operator valued vector measures \(T: \Sigma\to L(H,K)\), defined on a \(\sigma\)-algebra \(\Sigma\), in the case of two Hilbert spaces H, K. In fact, they show that the 2-majorizability characterization of dilatable vector measures \(\mu\) : \(\Sigma\to H\), obtained by the reviewer, is the property which gives a necessary and sufficient condition for an operator valued measure \(T: \Sigma\to L(H,K)\) to admit a dilation of the form (*) \(T(\Delta)=SE(\Delta)R\), \(\Delta\in \Sigma\), where E(\(\cdot)\) is a spectral measure on a Hilbert space \(K_ 0\) and \(R\in L(H,K_ 0)\), \(S\in L(K_ 0,K).\)
Among other results, the authors then present sufficient conditions on T for the existence of (*). A counterexample is given showing that the dilation (*) is not possible for all operator valued measures \(T: \Sigma\to L(H,K).\)
At the end of the paper the dilation result (*) is applied to obtain results on stationary dilations of Hilbert space-valued harmonizable and, especially, V-bounded stochastic processes, extending some of the results on the dilation of scalar valued harmonizable stochastic processes [cf. A. G. Miamee and H. Salehi, Indiana Univ. Math. J. 27, 37-50 (1978; Zbl 0353.60036)].
Reviewer: H.Niemi

MSC:
60G10 Stationary stochastic processes
47A20 Dilations, extensions, compressions of linear operators
60G12 General second-order stochastic processes
46G10 Vector-valued measures and integration
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