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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the existence of invariant subspaces in spaces with indefinite metric

Author: Kyûya Masuda
Journal: Proc. Amer. Math. Soc. 32 (1972), 440-444
MSC: Primary 47A15; Secondary 46D05, 47B50
MathSciNet review: 0295122
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Abstract: Let $ {P_1},{P_2}$ be complementary projections in Hilbert space H. Let U be a one-to-one and onto operator in H with $ Q(Ux) = Q(x)$, where $ Q(x) = {\left\Vert {{P_1}x} \right\Vert^2} - {\left\Vert {{P_2}x} \right\Vert^2}$. The sufficient condition is given for the unique existence of maximal subspace L invariant under all operators commuting with U, and such that $ Q(x) \geqq 0,x \in L$. The result was obtained in the course of attacking the problem proposed by Phillips [5] (see also [1]).

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  • [1] J. William Helton, Unitary operators on a space with an indefinite inner product, J. Functional Analysis 6 (1970), 412–440. MR 0415397
  • [2] M. G. Kreĭn, A new application of the fixed-point principle in the theory of operators in a space with indefinite metric, Dokl. Akad. Nauk SSSR 154 (1964), 1023–1026 (Russian). MR 0169059
  • [3] Heinz Langer, Eine Verallgemeinerung eines Satzes von L. S. Pontrjagin, Math. Ann. 152 (1963), 434–436 (German). MR 0158266,
  • [4] M. A. Naĭmark, On commuting unitary operators in spaces with indefinite metric, Acta Sci. Math. (Szeged) 24 (1963), 177–189. MR 0161158
  • [5] R. S. Phillips, On dissipative operators, Lecture Series in Differential Equations, Van Nostrand, Princeton, N.J., 1969.

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Keywords: Maximal positive invariant subspace, indefinite metric space
Article copyright: © Copyright 1972 American Mathematical Society