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On a certain class of nonstationary sequences in Hilbert space. (English) Zbl 1031.47008
In the present paper, the functions of correlation \(K(n,m):=\langle X(n),X(m)\rangle\), where \(n,m\in \mathbb{Z},\) is studied in Hilbert space \(H\) for certain sequences \(X(n).\) A. N. Kolmogorov [Bull. Mosk. Gos. Univ. Mat. 2, 1-40 (1941; Zbl 0063.03291)] showed that if \(X(n)\) is stationary (\(K(n,m)=K(n-m)\)) and \(n\in \mathbb{Z},\) then \(X(n)=U^nx_0\), \(x_0=X(0),\) where \(U\) is a unitary operator acting in the space \(H_X\) which is defined as the closed linear envelope of \(X=\{X(n)\mid n\in \mathbb{Z}\}.\) In this case, \(K(n,m)=\int_{-\pi}^{+\pi}e^{i(n-m)\lambda}dF_X(\lambda),\) where \(F_X\) is spectral function of \(X(n).\)
In this paper, nonstationary sequences of the form \(X(n)=T^nx_0\), \(x_0\in H,\) are studied, where \(T\) is a linear contraction (\(||T||\leq 1\)) in \(H.\) The general form of \(K(n,m)\) and the asymptotic behaviour \(\lim_{p\to+\infty}K(n+p,m+p)\) are given in the paper using the spectral methods of nonunitary operators.
MSC:
47A45 Canonical models for contractions and nonselfadjoint linear operators
60G12 General second-order stochastic processes
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