On the Hamiltonian property of linear dynamical systems in Hilbert space

Conditions for the operator differential equation x˙=Ax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot x = Ax$$\end{document} 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.