Application of Banach limits to invariant measures of infinite-dimensional Hamiltonian flows

Applying an invariant measure on phase space, we study the Koopman representation of a group of symplectomorphisms in an infinite-dimensional Hilbert space equipped with a translation-invariant symplectic form. The phase space is equipped with a finitely additive measure, invariant under the group of symplectomorphisms generated by Liouville-integrable Hamiltonian systems. We construct an invariant measure of Lebesgue type by applying a special countable product of Lebesgue measures on real lines. An invariant measure of Banach type is constructed by applying a countable product of Banach measures (defined by the Banach limit) on real lines. One of the advantages of an invariant measure of Banach type compared to an invariant measure of Lebesgue type is finiteness of the values of this measure in the entire space. The introduced invariant measures help us to describe both the strong continuity subspaces of the Koopman unitary representation of an infinite-dimensional Hamiltonian flow and the spectral properties of the constraint generator of the unitary representation on the invariant strong continuity subspace.