Invariant measures of infinite dimensional Hamiltonian systems and properties of Koopman groups

We study the class of finite additive measures on the real separable Hilbert space E endowed with a shift-invariant symplectic form. Any measure from this class is invariant with respect to the group of symplectomorphisms preserving two-dimensional symplectic subspaces (see [1]). A Hamiltonian flow in a real separable Hilbert space preserving two-dimensional symplectic subspaces is presented by the Koopman unitary group in the space of functions that are quadratically integrable by invariant measure. The spectrum of the generator of a Koopman group is described. The invariant subspace of strong continuity of Koopman group is determined. References [1] Glazatov V. A., Sakbaev V. Zh. Measures on Hilbert space invariant with respect to Hamiltonian flows // Ufa Math. J. 2022. Vol. 14. No 2. P.3—21