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Unconditional ideals of finite rank operators

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Abstract

Let X be a Banach space. We give characterizations of when \( \mathcal{F}(Y,X) \) is a u-ideal in \( \mathcal{W}(Y,X) \) for every Banach space Y in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when \( \mathcal{F}(X,Y) \) is a u-ideal in \( \mathcal{W}(X,Y) \) for every Banach space Y, when \( \mathcal{F}(Y,X) \) is a u-ideal in \( \mathcal{W}(Y,X^{ * * } ) \) for every Banach space Y, and when \( \mathcal{F}(Y,X) \) is a u-ideal in \( \mathcal{K}(Y,X^{ * * } ) \) for every Banach space Y.

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References

  1. P. K. Belobrov: Minimal extension of linear functionals onto the second conjugate space. Mat. Zametki 27 (1980), 439–445, 494.

    MATH  MathSciNet  Google Scholar 

  2. P. G. Casazza and N. J. Kalton: Notes on approximation properties in separable Banach spaces. Geometry of Banach Spaces, Proc. Conf. Strobl 1989, London Mathematical Society Lecture Note Series 158 (P.F.X. Müller and W. Schachermayer, eds.), Cambridge University Press, 1990, pp. 49–63.

  3. W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński: Factoring weakly compact operators. J. Functional Analysis 17 (1974), 311–327.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Feder and P. Saphar: Spaces of compact operators and their dual spaces. Israel J. Math. 21 (1975), 38–49.

    Article  MATH  MathSciNet  Google Scholar 

  5. G. Godefroy, N. J. Kalton and P. D. Saphar: Unconditional ideals in Banach spaces. Studia Math. 104 (1993), 13–59.

    MATH  MathSciNet  Google Scholar 

  6. G. Godefroy and P. Saphar: Duality in spaces of operators and smooth norms on Banach spaces. Illinois J. Math. 32 (1988), 672–695.

    MATH  MathSciNet  Google Scholar 

  7. P. Harmand, D. Werner and W. Werner: M-ideals in Banach Spaces and Banach Algebras. Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993.

    MATH  Google Scholar 

  8. J. Johnson and J. Wolfe: On the norm of the canonical projection of E*** onto E⊥. Proc. Amer. Math. Soc. 75 (1979), 50–52.

    Article  MATH  MathSciNet  Google Scholar 

  9. N. J. Kalton: Locally complemented subspaces and \( \mathcal{L}_p \)-spaces for 0 < p < 1. Math. Nachr. 115 (1984), 71–97.

    Article  MATH  MathSciNet  Google Scholar 

  10. V. Lima: The weak metric approximation property and ideals of operators. J. Math. Anal. A. 334 (2007), 593–603.

    Article  MATH  MathSciNet  Google Scholar 

  11. V. Lima and Å. Lima: Geometry of spaces of compact operators. Arkiv für Matematik 46 (2008), 113–142.

    Article  MATH  MathSciNet  Google Scholar 

  12. V. Lima and Å. Lima: Ideals of operators and the metric approximation property. J. Funct. Anal. 210 (2004), 148–170.

    Article  MATH  MathSciNet  Google Scholar 

  13. Å. Lima: Intersection properties of balls and subspaces in Banach spaces. Trans. Amer. Math. Soc. 227 (1977), 1–62.

    Article  MATH  MathSciNet  Google Scholar 

  14. Å. Lima: The metric approximation property, norm-one projections and intersection properties of balls. Israel J. Math. 84 (1993), 451–475.

    Article  MATH  MathSciNet  Google Scholar 

  15. Å. Lima: Property (wM*) and the unconditional metric compact approximation property. Studia Math. 113 (1995), 249–263.

    MATH  MathSciNet  Google Scholar 

  16. Å. Lima, O. Nygaard and E. Oja: Isometric factorization of weakly compact operators and the approximation property. Israel J. Math. 119 (2000), 325–348.

    Article  MATH  MathSciNet  Google Scholar 

  17. Å. Lima and E. Oja: Ideals of finite rank operators, intersection properties of balls, and the approximation property. Studia Math. 133 (1999), 175–186.

    MATH  MathSciNet  Google Scholar 

  18. Å. Lima and E. Oja: Hahn-Banach extension operators and spaces of operators. Proc. Amer. Math. Soc. 130 (2002), 3631–3640 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  19. Å. Lima and E. Oja: Ideals of compact operators. J. Aust. Math. Soc. 77 (2004), 91–110.

    Article  MATH  MathSciNet  Google Scholar 

  20. Å. Lima and E. Oja: Ideals of operators, approximability in the strong operator topology, and the approximation property. Michigan Math. J. 52 (2004), 253–265.

    Article  MATH  MathSciNet  Google Scholar 

  21. Å. Lima and E. Oja: The weak metric approximation property. Math. Ann. 333 (2005), 471–484.

    Article  MATH  MathSciNet  Google Scholar 

  22. Å. Lima, E. Oja, T. S. S. R. K. Rao and D. Werner: Geometry of operator spaces. Michigan Math. J. 41 (1994), 473–490.

    Article  MATH  MathSciNet  Google Scholar 

  23. J. Lindenstrauss and L. Tzafriri: Classical Banach Spaces I. Springer, Berlin-Heidelberg-New York, 1977.

    MATH  Google Scholar 

  24. E. Oja: Uniqueness of the extension of linear continuous functionals according to the Hahn-Banach theorem. Eesti NSV Tead. Akad. Toimetised Füüs.-Mat. 33 (1984), 424–438, 473. (In Russian.)

    MathSciNet  Google Scholar 

  25. E. Oja: Strong uniqueness of the extension of linear continuous functionals according to the Hahn-Banach theorem. Mat. Zametki 43 (1988), 237–246, 302 (In Russian.); English translation in Math. Notes 43 (1988), 134–139.

    MathSciNet  Google Scholar 

  26. E. Oja: Dual de l’espace des opérateurs linéaires continus. C. R. Acad. Sc. Paris, Sér. A 309 (1989), 983–986.

    MATH  MathSciNet  Google Scholar 

  27. E. Oja: Extension of functionals and the structure of the space of continuous linear operators. Tartu. Gos. Univ., Tartu (1991). (In Russian.)

  28. E. Oja: HB-subspaces and Godun sets of subspaces in Banach spaces. Mathematika 44 (1997), 120–132.

    Article  MATH  MathSciNet  Google Scholar 

  29. E. Oja: Geometry of Banach spaces having shrinking approximations of the identity. Trans. Amer. Math. Soc. 352 (2000), 2801–2823.

    Article  MATH  MathSciNet  Google Scholar 

  30. E. Oja: Operators that are nuclear whenever they are nuclear for a larger range space. Proc. Edinb. Math. Soc. 47 (2004), 679–694.

    Article  MATH  MathSciNet  Google Scholar 

  31. E. Oja: The impact of the Radon-Nikodým property on the weak bounded approximation property. Rev. R. Acad. Cien. Serie A. Mat. 100 (2006), 325–331.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Agder, Servicebox 422, 4604, Kristiansand, Norway

    Trond A. Abrahamsen & Åsvald Lima

  2. Mathematics Department, University of Missouri, 202 Mathematical Sciences Bldg, Columbia, MO, 652 11, USA

    Vegard Lima

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  1. Trond A. Abrahamsen
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  2. Åsvald Lima
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  3. Vegard Lima
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Correspondence to Trond A. Abrahamsen.

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Abrahamsen, T.A., Lima, Å. & Lima, V. Unconditional ideals of finite rank operators. Czech Math J 58, 1257–1278 (2008). https://doi.org/10.1007/s10587-008-0085-9

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