On the Hamiltonian property of linear dynamical systems in Hilbert space

详细

Conditions for the operator differential equation \(\dot x = Ax\) possessing a quadratic first integral (1/2)(Bx, x) to be Hamiltonian are obtained. In the finite-dimensional case, it suffices to require that ker B ⊂ ker A*. For a bounded linear mapping x → Ωx possessing a first integral, sufficient conditions for the preservation of the (possibly degenerate) Poisson bracket are obtained.