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Acta Applicandae Mathematica

, Volume 27, Issue 1–2, pp 1–22 | Cite as

Positive operators on Krein spaces

  • Y. A. Abramovich
  • C. D. Aliprantis
  • O. Burkinshaw
Article

Abstract

A Krein operator is a positive operator, acting on a partially ordered Banach space, that carries positive elements to strong units. The purpose of this paper is to present a survey of the remarkable spectral properties (most of which were established by M.G. Krein) of these operators. The proofs presented here seem to be simpler than the ones existing in the literature. Some new results are also obtained. For instance, it is shown that every positive operator on a Krein space which is not a multiple of the identity operator has a nontrivial hyperinvariant subspace.

Mathematics Subject Classifications (1991)

46C50 47B65 47B37 

Key words

partially ordered Banach space Krein space Krein operator hyperinvariant subspace 

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References

  1. 1.
    I.A. Bakhtin, On the existence of eigenvectors of linear positive noncompact operators, Math. USSR-Sb. (N.S.) 64 (1964), pp. 102–114 (in Russian).Google Scholar
  2. 2.
    I.A. Bakhtin, On the existence of common eigenvector for a commutative family of linear positive operators, Math. USSR-Sb. (N.S.) 67 (1965), pp. 267–278 (in Russian).Google Scholar
  3. 3.
    I.A. Bakhtin, M.A. Krasnoselsky and V.Ya. Stezenko, Continuity of linear positive operators, Sibirsk. Math. Ž. 3 (1962), pp. 156–160.Google Scholar
  4. 4.
    G. Frobenius, Über Matrizen aus nicht-negativen Elementen, Sitz. Berichte Kgl. Preuß. Akad. Wiss. Berlin (1912), pp. 456–477.Google Scholar
  5. 5.
    G. Jameson, Ordered Linear Spaces, Lecture Notes in Mathematics, 141, Springer-Verlag, Heidelberg, 1970.Google Scholar
  6. 6.
    A.K. Kitover, The spectral properties of weighted homomorphisms in algebras of continuous functions and their applications, Zap. Nauĉn. Sem. Leningrad Otdel. Mat. Inst. Steklov. (LOMI) 107 (1982), pp. 89–103.Google Scholar
  7. 7.
    M.G. Krein, Fundamental properties of normal conical sets in a Banach space, Dokl. Akad. Nauk USSR 28 (1940), pp. 13–17.Google Scholar
  8. 8.
    M.G. Krein and M.A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk 3 (1948), pp. 3–95 (in Russian). Also, Amer. Math. Soc. Transl. 26 (1950).Google Scholar
  9. 9.
    A.A. Markov, Some theorems on Abelian collections, Dokl. Akad. Nauk USSR 10 (1936), pp. 311–313.Google Scholar
  10. 10.
    L. Nachbin, On the continuity of positive linear transformations, Proc. Internat. Congress of Math. (1950), pp. 464–465.Google Scholar
  11. 11.
    I. Namioka, Partially ordered linear topological spaces, Mem. Amer. Math. Soc. 24 (1957).Google Scholar
  12. 12.
    A.L. Peressini, Ordered Topological Vector Spaces, Harper & Row, New York, 1967.Google Scholar
  13. 13.
    O. Perron, Zur Theorie der Matrizen, Math. Ann. 64 (1907), pp. 248–263.Google Scholar
  14. 14.
    H.H. Schaefer, Topological Vector Spaces, Springer-Verlag, Berlin-New York, 1971.Google Scholar
  15. 15.
    B.Z. Vulikh, Introduction in the Theory of Cones in Normed Spaces, Kalinin State University, 1977.Google Scholar
  16. 16.
    B.Z. Vulikh, Special Topics in Geometry of Cones in Normed Spaces, Kalinin State University, 1978.Google Scholar
  17. 17.
    Y.C. Wong and K. F. Ng, Partially Ordered Topological Vector Spaces, Clarendon Press, Oxford, 1973.Google Scholar
  18. 18.
    A.C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam, 1983.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • Y. A. Abramovich
    • 1
  • C. D. Aliprantis
    • 1
  • O. Burkinshaw
    • 1
  1. 1.Department of MathematicsIUPUIIndianapolisU.S.A

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