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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Linear dynamical systems

Author: E. Flytzanis
Journal: Proc. Amer. Math. Soc. 55 (1976), 367-370
MSC: Primary 28A65
MathSciNet review: 0407236
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Abstract: We consider a probability measure $ m$ on a Hilbert space $ X$ and a bounded linear transformation on $ X$ that preserves the measure. We characterize the linear dynamical systems $ (X,m,T)$ for the cases where either $ X$ is finite dimensional or $ T$ is unitary and we give an example where $ T$ is a weighted shift operator. We apply the results to the limit identification problem for a vector-valued ergodic theorem of A. Beck and J. T. Schwartz, $ {n^{ - 1}}(\Sigma _i^n{T^i}{F_i}) \to \overline F $ a.s., where $ {F_i}$ is a stationary sequence of integrable $ X$-valued random variables and $ T$ a unitary operator on $ X$.

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Keywords: Measure preserving transformation, dynamical system, spectrum, linear operator Hilbert space measure, ergodic theorem, stationary stochastic process, shift operator
Article copyright: © Copyright 1976 American Mathematical Society