Skip to main content

Geometry of spaces of compact operators


For non-reflexive Banach spaces X, Y, for a very smooth point in the space of compact linear operators K(X, Y), we give several sufficient conditions for the adjoint to be a very smooth point in K(Y *,X*). We exhibit a new class of extreme points in the dual unit ball of injective product spaces. These ideas are also related to Birkhoff-James orthogonality in spaces of operators.

This is a preview of subscription content, access via your institution.


  1. E. Behrends, M-structure and the Banach-Stone theorem, Lecture Notes in Mathematics 736, Springer, Berlin, 1979.

    Book  Google Scholar 

  2. M. Cambern and P. Greim, The bidual of C(X,E), Proc. Amer. Math. Soc., 85 (1982), 53–58.

    MathSciNet  MATH  Google Scholar 

  3. M. Cambern and P. Greim, Uniqueness of preduals for spaces of continuous vector functions, Canad. Math. Bull., 32 (1989), 98–104.

    Article  MathSciNet  Google Scholar 

  4. J. Diestel and J. J. Uhl, Vector measures, Mathematical Surveys 15, Amer. Math. Soc., Providence, R.I., 1977.

    Book  Google Scholar 

  5. G. Godefroy and V. Indumathi, Norm-to-weak upper semi-continuity of the duality and pre-duality mappings, Set-Valued Anal., 10 (2002), 317–330.

    Article  MathSciNet  Google Scholar 

  6. P. Harmand, D. Werner and W. Werner, M-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics 1547, Springer-Verlag, Berlin, 1993.

    Book  Google Scholar 

  7. S. Heinrich, The differentiability of the norm in spaces of operators, Funkcional. Anal. i Prilozen., 9 (1975), 93–94 (in Russian); English translation: Funct. Anal. Appl. (4), 9 (1975), 360–362.

    Article  MathSciNet  Google Scholar 

  8. R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Pitman, 1993.

    MATH  Google Scholar 

  9. R. B. Holmes Geometric functional analysis and its applications, Graduate Texts in Math. 24, Springer-Verlag, New York - Heidelberg, 1975.

    Book  Google Scholar 

  10. Z. Hu and B. L. Lin, RNP and CPCP in Lebesgue-Bochner function spaces, Illinois J. Math., 37 (1993), 328–347.

    Article  MathSciNet  Google Scholar 

  11. B. L. Lin P. K. Lin and S. L. Troyanski, Characterizations of denting points, Proc. Amer. Math. Soc., 102 (1988), 526–528.

    Article  MathSciNet  Google Scholar 

  12. T. S. S. R. K. Rao L(X, C(K)) as a dual space, Proc. Amer. Math. Soc., 110 (1990), 727–729.

    MathSciNet  MATH  Google Scholar 

  13. T. S. S. R. K. Rao On ideals in Banach spaces, Rocky Mountain J. Math., 31 (2001), 595–609.

    Article  MathSciNet  Google Scholar 

  14. T. S. S. R. K. Rao Smooth points in spaces of operators, Linear Algebra Appl., 517 (2017), 129–133.

    Article  MathSciNet  Google Scholar 

  15. W. Ruess and C. Stegall, Extreme points in duals of operator spaces, Math. Ann., 261 (1982), 535–546.

    Article  MathSciNet  Google Scholar 

  16. F. Sullivan, Geometric properties determined by the higher duals of a Banach space, Illinois J. Math., 21 (1977), 315–331.

    Article  MathSciNet  Google Scholar 

  17. P. Wöjcik, Orthogonality of compact operators, Expo. Math., 35 (2017), 86–94.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to T. S. S. R. K. Rao.

Additional information

Communicated by L. Molnár

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rao, T.S.S.R.K. Geometry of spaces of compact operators. ActaSci.Math. 85, 495–505 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

AMS Subject Classification (2000)

  • 47L05
  • 46B28
  • 46B25

Key words and phrases

  • smooth points
  • very smooth points
  • adjoints of operators
  • spaces of operators
  • essential norm
  • injective and projective tensor product spaces