Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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HEREDITARY PROPERTIES OF CERTAIN IDEALS OF COMPACT OPERATORS

  • Published : 2004.08.01
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Abstract

Let X be a Banach space and Z a closed subspace of a Banach space Y. Denote by Ｌ(X, Y) the space of all bounded linear operators from X to Y and by Ｋ(X, Y) its subspace of compact linear operators. Using Hahn-Banach extension operators corresponding to ideal projections, we prove that if either $X^{**}$ or $Y^{*}$ has the Radon-Nikodym property and Ｋ(X, Y) is an M-ideal (resp. an HB-subspace) in Ｌ(X, Y), then Ｋ(X, Z) is also an M-ideal (resp. HB-subspace) in Ｌ(X, Z). If Ｌ(X, Y) has property SU instead of being an M-ideal in Ｌ(X, Y) in the above, then Ｋ(X, Z) also has property SU in Ｌ(X, Z). If X is a Banach space such that $X^{*}$ has the metric compact approximation property with adjoint operators, then M-ideal (resp. HB-subspace) property of Ｋ(X, Y) in Ｌ(X, Y) is inherited to Ｋ(X, Z) in Ｌ(X, Z).

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References

  1. E. M. Alfsen and E. G. Effros, Structure in real Banach spaces, Ann. of Math. 96 (1972), 98–173
  2. P. G. Casazza and N. J. Kalton, Notes on approximation properties in separable Banach spaces, In Geometry of Banach spaces, Proc. Conf. Strobl. (1989) (eds. P.F.X. Muller and W. Schachermayer). London Math. Soc. Lecture Note Ser. 158 (1990), 49–63
  3. M. Feder and P. Saphar, Spaces of compact operators and their dual spaces, Israel J. Math. 21 (1975), 38–49
  4. G. Godefroy, N. J. Kalton and P. D. Saphar, Unconditional ideals in Banach spaces, IBID Press. 104 (1993), 13–59
  5. P. Harmand, D. Werner and W. Werner, M-ideals in Banach spaces and Banach Algebras, Lecture Notes in Math. 1547 (Springer Berlin-Heiderberg-New York 1993)
  6. J. Hennefeld, M-ideals, HB-subspace, and compact operator, Indiana Univ. Math. J. 28 (1979), 927–934
  7. J. Johnson, Remarks on Banach spaces of compact operators, J. Funct. Anal. 32 (1979), 304–311
  8. A. Lima, Uniqueness of Hahn-Banach extensions and liftings of linear dependences, Math. Scand. 53 (1983), 97–113
  9. A. Lima, O. Nygaard and E. Oja, Isometric factorization of weakly compact operators and the approximation property, Israel J. Math. 119 (2000), 325–348 https://doi.org/10.1007/BF02810673
  10. A. Lima and E. Oja, Ideals of finite rank operators, intersection properties of balls, and the approximation property, Studia Math. 133 (1999), no. 2, 175–186
  11. A. Lima and E. Oja, Ideals of compact operators, preprint
  12. A. Lima and E. Oja, Hahn-Banach extension operators and spaces of operators, Proc. Amer. Math. Soc. 130 (2002), 3631–3640
  13. A. Lima and E. Oja, Ideals of operators, approximality in the strong operator toporogy, and the approximation property, preprint
  14. A. Lima, E. Oja, T.S.S.R.K. Rao and D. Werner, Geometry of operator spaces, Michigan Math. J. 41 (1994), 473–490
  15. E. F. Oja, Strong uniqueness of the extension of linear continuous functionals according to the Hahn-Banach theorem, Mat. Zametki 43 (1988), 237–46 (in Russian) Math. Notes 43 (1988), 134–139
  16. E. F. Oja, HB-subspaces and Godun sets of subspaces in Banach spaces, Matematika 44 (1997), 120–132
  17. R. Phelps, Uniqueness of Hahn-Banach extensions and unique best approximation, Trans. Amer. Math. Soc. 95 (1960), 238–255