Linear dynamical systems with quadratic invariant in Hilbert space

In the talk problems related to linear differential equations and linear maps in Hilbert spaces admitting a quadratic invariant will be discussed. Existence of a quadratic invariant allows to introduce a natural simplectic structure and allows to investigate problems of existence of invariant measures, of the Hamiltonian property of linear systems and their complete integrability. General results will be illustrated by evolution equations of mathematical physics. In particular simplectic geometry of the Koopman operator in ergodic theory will be discussed.