Linear dynamical systems
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- by E. Flytzanis PDF
- Proc. Amer. Math. Soc. 55 (1976), 367-370 Request permission
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 $X$.References
- Anatole Beck and J. T. Schwartz, A vector-valued random ergodic theorem, Proc. Amer. Math. Soc. 8 (1957), 1049–1059. MR 98162, DOI 10.1090/S0002-9939-1957-0098162-6
- K. Jacobs, Lecture notes on ergodic theory, 1962/63. Parts I, II, Aarhus Universitet, Matematisk Institut, Aarhus, 1963. MR 0159922
- Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
Additional Information
- © Copyright 1976 American Mathematical Society
- 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