Linear dynamical systems

Author:
E. Flytzanis

Journal:
Proc. Amer. Math. Soc. **55** (1976), 367-370

MSC:
Primary 28A65

DOI:
https://doi.org/10.1090/S0002-9939-1976-0407236-0

MathSciNet review:
0407236

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Abstract: We consider a probability measure on a Hilbert space and a bounded linear transformation on that preserves the measure. We characterize the linear dynamical systems for the cases where either is finite dimensional or is unitary and we give an example where 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, a.s., where is a stationary sequence of integrable -valued random variables and a unitary operator on .

**[1]**A. Beck and J. T. Schwartz,*A vector-valued ergodic theorem*, Proc. Amer. Math. Soc.**8**(1957), 1049-1059. MR**20**#4624. MR**0098162 (20:4624)****[2]**K. Jacobs,*Lecture notes on ergodic theory*. Parts I, II, Mathematisk Institut, Aarhus Universitet, Aarhus, 1963. MR**28**#3138; erratum,**28**, p. 1247. MR**0159922 (28:3138)****[3]**P. R. Halmos,*A Hilbert space problem book*, Van Nostrand, Princeton, N. J., 1967. MR**34**#8178. MR**0208368 (34:8178)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0407236-0

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