Hamiltonsche Bahnen ohne Zerspaltungseigenschaft. Die Loesung einer Aufgabe von M. G. Krein

Abstract

There 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?