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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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