On the Koopman representation of Hamiltonian flows in infinite dimensional spaces with invariant measure

We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on a Hilbert space that are invariant with respect to flows of completely integrable Hamiltonian systems and allow one to describe Hamiltonian flows in a phase space in terms of unitary groups in the space of functions squarely integrable with respect to an invariant measure are studied. The properties of the Koopman representation are described using the example of the Hamiltonian of a countable set of noninteracting harmonic oscillators. A spectral analysis of the generator of Koopman group for such a Hamiltonian is carried out. An invariant subspace of strong continuity of Koopman unitary group is described in terms of spectrum of the generator.