Arkiv för Matematik

  • Ark. Mat.
  • Volume 41, Number 2 (2003), 233-252.

A property of strictly singular one-to-one operators

George Androulakis and Per Enflo

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Abstract

We prove that if T is a strictly singular one-to-one operator defined on an infinite dimensional Banach space X, then for every infinite dimensional subspace Y of X there exists an infinite dimensional subspace Z of X such that Z∩Y is infinite dimensional, Z contains orbits of T of every finite length and the restriction of T to Z is a compact operator.

Note

The research was partially supported by NSF.

Article information

Source
Ark. Mat., Volume 41, Number 2 (2003), 233-252.

Dates
Received: 7 January 2002
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898803

Digital Object Identifier
doi:10.1007/BF02390813

Mathematical Reviews number (MathSciNet)
MR2011919

Zentralblatt MATH identifier
1077.47019

Rights
2003 © Institut Mittag-Leffler

Citation

Androulakis, George; Enflo, Per. A property of strictly singular one-to-one operators. Ark. Mat. 41 (2003), no. 2, 233--252. doi:10.1007/BF02390813. https://projecteuclid.org/euclid.afm/1485898803


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References

  • Androulakis, G., Odell, E., Schlumprecht, T. and Tomczak-Jaegermann, N., On the structure of the spreading models of a Banach space, Preprint, 2002.
  • Androulakis, G. and Schlumprecht, T., Strictly singular non-compact operators exist on the space of Gowers-Maurey, J. London Math. Soc. 64 (2001), 655–674.
  • Aronszajn, N. and Smith, K. T., Invariant subspaces of completely continuous operators, Ann. of Math. 60 (1954), 345–350.
  • Enflo, P., On the invariant subspace problem in Banach spaces, in Seminaire Maurey-Schwartz (1975–1976). Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. 14–15, Centre Math. École Polytechnique, Palaiseau, 1976.
  • Enflo, P., On the invariant subspace problem for Banach spaces, Acta Math. 158 (1987), 213–313.
  • Ferenczi, V., A uniformly convex hereditarily indecomposable Banach space, Israel J. Math. 102 (1997), 199–225.
  • Gasparis, I., Strictly singular non-compact operators on hereditarily indecomposable Banach spaces, Proc. Amer. Math. Soc. 131 (2003), 1181–1189.
  • Gowers, W. T., A remark about the scalar-plus-compact problem, in Convex Geometric Analysis (Berkeley, Calif., 1996). (Ball, K. M. and Milman, V., eds.), Math. Sci. Res. Inst. Publ. 34, pp. 111–115, Cambridge Univ. Press, Cambridge, 1999.
  • Gowers, W. T. and Maurey, B., The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851–874.
  • Lomonosov, V. I., Invariant subspaces of the family of operators that commute with a completely continuous operator, Funktsional. Anal. i Prilozhen. 7:3 (1973), 55–56 (Russian). English transl.: Funct. Anal. Appl. 7 (1973), 213–214.
  • Read, C. J., A solution to the invariant subspace problem, Bull. London Math. Soc. 16 (1984), 337–401.
  • Read, C. J., A solution to the invariant subspace problem on the space l1, Bull. London Math. Soc. 17 (1985), 305–317.
  • Read, C. J., A short proof concerning the invariant subspace problem, J. London Math. Soc. 34 (1986), 335–348.
  • Read, C. J., Strictly singular operators and the invariant subspace problem, Studia Math. 132 (1999), 203–226.