# On the Hamiltonian property of linear dynamical systems in Hilbert space

Authors
• 1 Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia , Moscow (Russia)
• 2 Lomonosov Moscow State University, Moscow, Russia , Moscow (Russia)
Type
Published Article
Journal
Mathematical Notes
Publisher
Publication Date
May 01, 2017
Volume
101
Issue
5-6
Pages
1033–1039
Identifiers
DOI: 10.1134/S0001434617050303
Source
Springer Nature
Keywords
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.