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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the existence of invariant subspaces in spaces with indefinite metric
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by Kyûya Masuda PDF
Proc. Amer. Math. Soc. 32 (1972), 440-444 Request permission

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 $Q(x) = {\left \| {{P_1}x} \right \|^2} - {\left \| {{P_2}x} \right \|^2}$. 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]).
References
  • J. William Helton, Unitary operators on a space with an indefinite inner product, J. Functional Analysis 6 (1970), 412–440. MR 0415397, DOI 10.1016/0022-1236(70)90070-4
  • 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
  • Heinz Langer, Eine Verallgemeinerung eines Satzes von L. S. Pontrjagin, Math. Ann. 152 (1963), 434–436 (German). MR 158266, DOI 10.1007/BF01470908
  • M. A. Naĭmark, On commuting unitary operators in spaces with indefinite metric, Acta Sci. Math. (Szeged) 24 (1963), 177–189. MR 161158
  • R. S. Phillips, On dissipative operators, Lecture Series in Differential Equations, Van Nostrand, Princeton, N.J., 1969.
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • 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