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Some aspects of the spectral theory of positive operators

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Abstract

We consider various aspects of the following problem: Let T be a positive operator on a Banach lattice such that σ(T)={1}. Does it follow that T≥1?

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References

  1. C.A. Akemann, P.A. Ostrand, The spectrum of a derivation of a C *-algebra, J. London Math. Soc. (2) 13 (1976), pp. 525–530.

    Google Scholar 

  2. W. Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. J. 32 (1983), pp. 199–215.

    Google Scholar 

  3. W. Arendt, D.R. Hart, The spectrum of quasi-invertible disjointness preserving operators, J. Funct. 68 (1986), pp. 149–167.

    Google Scholar 

  4. A. Atzmon, Operators which are annihilated by analytic functions and invariant subspaces, Acta Math. 144 (1980), pp. 27–63.

    Google Scholar 

  5. B. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North-Holland, Amsterdam, 1988.

    Google Scholar 

  6. R.P. Boas, Jr., Entire Functions, Academic Press, 1954.

  7. J. Esterle, Quasimultipliers, representations on H , and the closed ideal problem for commutative Banach algebras, in: Radical Banach Algebras and Automatic Continuity, Lecture Notes in Math. 975 pp. 66–162.

  8. C.B. Huijsmans, Elements with unit spectrum in Banach lattice algebras, Indag. Math. 50 (1988), pp. 43–45.

    Google Scholar 

  9. C.B. Huijsmans, An elementary proof of a theorem of Schaefer, Wolff and Arendt, Proc. Amer. Math. Soc. 105 (1989), pp. 632–635.

    Google Scholar 

  10. From A to Z, Proceeding of a Symposium in Honour of A.C. Zaanen: Edited by C.B. Huijsmans, M.A. Kaashoek, W.A.J. Luxemburg and W.K. Vietsch: Mathematical Center Tracts 149, Amsterdam, 1982.

  11. B. Johnson, Automorphisms of commutative Banach algebras, Proc. Amer. Math. Soc. 40 (1973), pp. 497–499.

    Google Scholar 

  12. H. Kamowitz, S. Scheinberg, The spectrum of automorphisms of Banach algebras, J. Funct. Analysis 4 (1969), pp. 268–276.

    Google Scholar 

  13. Y. Katznelson, L. Tzafriri, On power-bounded operators, J. Funct. Analysis 68 (1986), pp. 313–328.

    Google Scholar 

  14. W.A.J. Luxemburg, Some aspects of the theory of Riesz spaces, University of Arkansas, Lecture Notes in Mathematics, Fayetteville 4 (1979).

  15. W.A.J. Luxemburg, A.C. Zaanen, Riesz Spaces I, North-Holland, Amsterdam, 1971.

    Google Scholar 

  16. H.H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, 1974.

  17. H.H. Schaefer, On the 0-spectrum of order bounded operators, Math. Z. 154 (1977), pp. 78–84.

    Google Scholar 

  18. H.H. Schaefer, On positive contractions in L p-spaces, Trans. Amer. Math. Soc. 257 (1980), pp. 261–268.

    Google Scholar 

  19. H.H. Schaefer, M. Wolff, W. Arendt, On lattice isomorphisms and the groups of positive operators, Math. Z. 164 (1978), pp. 115–123.

    Google Scholar 

  20. A.R. Schep, Positive diagonal and triangular operators, J. Operator Theory 3 (1980), pp. 165–178.

    Google Scholar 

  21. J. Voigt, The Projection onto the center of operators in a Banach lattice, Math. Z. 199 (1988), pp. 115–117.

    Google Scholar 

  22. A.C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam, 1983.

    Google Scholar 

  23. X.D. Zhang, Two simple proofs for a theorem of Schaefer, Wolff and Arendt, Caltech, Pasadena, California, August, 1990. Submitted.

  24. X.D. Zhang, On Spectral properties of positive operators, Ph.D. Thesis, California Institute of Technology, Pasadena, California, USA (April, 1991).

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Zhang, XD. Some aspects of the spectral theory of positive operators. Acta Appl Math 27, 135–142 (1992). https://doi.org/10.1007/BF00046644

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