  In this work we consider hypercyclic operators as a special case of Polish
dynamical systems. In the first section we analyze the construction of 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 hypecyclic operators which, as Polish dynamical
systems are, M-systems, E-systems, and syndetically transitive systems.
