哈密尔顿铁路公司有Zerspaltungseigenschaft。这是M. G. Krein
Hamiltonsche Bahnen ohne Zerspaltungseigenschaft. Die Loesung einer Aufgabe von M. G. Krein
摘要：
构造线性Hamilton（动力）系统，使其非零轨线具有通常的渐近二分性。特别是有解决（在负）一个所谓的M。G.克林问题。定义：设J是周期为2的酉算子，U是线性算子.如果U^*JU = UJU^* = J，则U被称为J-酉的。KREIN问题：给定J-酉算子U，是否存在U-不变子空间L，比如，r（U| L）\leq 1？在算子U^*U-I是紧的特殊情况下，MG Krein在1964年给出了这个问题的正解。相反，我们将证明，在一般情况下，这样的子空间L不需要存在。进一步证明了定理：对每个实c>0，存在J-酉算子U使得（i）如果L是某个非零U-不变子空间，则r（U| L）>c;（ii）若L'是某个非零U^{-1}-不变子空间，则r（U^{-1}| L '）>c;这个结果既适用于实空间的情况，也适用于复空间的情况。此外，可以假设U是线性辛自同构。一个类似的结果是连续的情况下，动态的问题：是否存在一个非零的拟稳定的流形？
英文摘要：
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?
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