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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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