2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79035There are constructed linear Hamiltonian (dynamical) systems such that their no nonzero trajectory has usual asymptotical dichotomy property. In particular there is solved (in the negative) one of the so-called M. G. Krein problem. In fact Definition: Let J be period-2 unitary operator and U be linear operator. If U^*JU = UJU^* = J then U is said to be J-unitary. KREIN Problem: given a J-unitary operator U, does there exist an U-invariant subspace L, say, with r(U|L)\leq1 ? In the special case that the operator U^*U-I is compact this problem was solved in the positive by M.G.Krein in 1964. We shall show that, by contrast, in the general case such a subspace L needs not exist. Moreover, there asserts Theorem: For every real c>0 there exists some J-unitary operator U such that (i) if L is some nonzero U-invariant subspace, then r(U|L)>c; (ii) if L' is some nonzero U^{-1}-invariant subspace, then r(U^{-1}|L')>c; This result applies both to the real space case and to the complex space case. In addition, one can assume that U is linear symplectic automorphism. A similar result is obtained for the case of continuous 'dynamic' and for the question: does there exist a nonzero quasistable manifold?introduction, background and comments are extended, several propositions are added, Latex 2.09, in German, Journal-ref, e-mail: izvraen@ssu.samara.ru, http://www.mnu.samara.ru/~journalDynamical SystemsMathematical PhysicsRepresentation TheorySpectral Theorytext