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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
DOI: https://doi.org/10.1090/S0002-9939-1972-0295122-5
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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DOI: https://doi.org/10.1090/S0002-9939-1972-0295122-5
Keywords: Maximal positive invariant subspace, indefinite metric space
Article copyright: © Copyright 1972 American Mathematical Society