 0076 Sergej A. Choro\v savin
 Hamiltonsche Bahnen ohne Zerspaltungseigenschaft. Die Loesung einer
Aufgabe von M. G. Krein
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Feb 17, 00

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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 socalled M. G. Krein problem. In
fact
Definition: Let J be period2 unitary operator and U be linear operator.
If U^*JU = UJU^* = J then U is said to be Junitary.
KREIN Problem: given a Junitary operator U, does there exist an Uinvariant
subspace L, say, with r(UL)\leq1 ?
In the special case that the operator U^*UI 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 Junitary operator U such that
(i) if L is some nonzero Uinvariant subspace, then r(UL)>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?
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