Abstract
In this work we consider hypercyclic operators as a special case of Polish dynamical systems. In the first section we analyze a construction due to Bayart and Grivaux of a hypercyclic operator which preserves a Gaussian measure, and derive a description of the maximal spectral type of the Koopman operator associated to the corresponding measure preserving dynamical system. We then use this information to show the existence of a mildly but not strongly mixing hypercyclic operator on Hilbert space. In the last two sections we study hypercyclic and frequently hypercyclic operators which, as Polish dynamical systems, are M-systems, E-systems, and syndetically transitive systems.
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This research was supported by a grant of ISF 668/13
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Dayan, Y., Glasner, E. Hypercyclic operators, Gauss measures and Polish dynamical systems. Isr. J. Math. 208, 79–99 (2015). https://doi.org/10.1007/s11856-015-1194-4
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DOI: https://doi.org/10.1007/s11856-015-1194-4