Approximate Birkhoff–James orthogonality and smoothness in the space of bounded linear operators

  • Arpita Mal
  • Kallol PaulEmail author
  • T. S. S. R. K. Rao
  • Debmalya Sain


We study approximate Birkhoff–James orthogonality of bounded linear operators defined between normed linear spaces \(\mathbb {X}\) and \(\mathbb {Y}.\) As an application of the results obtained, we characterize smoothness of a bounded linear operator T under the condition that \(\mathbb {K}(\mathbb {X},\mathbb {Y}),\) the space of compact linear operators is an M-ideal in \(\mathbb {L}(\mathbb {X},\mathbb {Y}),\) the space of bounded linear operators.


Orthogonality Linear operators M-ideal L-ideal Smoothness 

Mathematics Subject Classification

Primary 46B28 Secondary 47L05 46B20 



  1. 1.
    Birkhoff, G.: Orthogonality in linear metric spaces. Duke Math. J. 1, 169–172 (1935)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chmieliński, J.: On an \(\epsilon \)-Birkhoff orthogonality. J. Inequal. Pure Appl. Math. 6(3), Article 79 (2005)Google Scholar
  3. 3.
    Chmieliński, J., Stypuła, T., Wójcik, P.: Approximate orthogonality in normed spaces and its applications. Linear Algebra Appl. 531, 305–317 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dragomir, S.S.: On approximation of continuous linear functionals in normed linear spaces. An. Univ. Timisoara Ser. Stiint. Mat. 29, 51–58 (1991)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Grza̧ślewicz, R., Younis, R.: Smooth points and M-ideals. J. Math. Anal. Appl. 175, 91–95 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Holmes, R.B.: Geometric Functional Analysis and Its Applications. Graduate Texts in Mathematics, vol. 24. Springer, New York (1975). x+246ppzbMATHGoogle Scholar
  7. 7.
    Harmand, P., Werner, D., Werner, W.: M-Ideals in Banach Spaces and Banach Algebras. Lecture Notes in Mathematics, vol. 1547. Springer, Berlin (1993)zbMATHGoogle Scholar
  8. 8.
    James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 61, 265–292 (1947)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Martin, M.: Norm-attaining compact operators. J. Funct. Anal. 267, 1585–1592 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mal, A., Sain, D., Paul, K.: On some geometric properties of operator spaces. Banach J. Math. Anal. 13(1), 174–191 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Paul, K., Sain, D., Mal, A.: Approximate Birkhoff–James orthogonality in the space of bounded linear operators. Linear Algebra Appl. 537, 348–357 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Paul, K., Sain, D., Ghosh, P.: Birkhoff–James orthogonality and smoothness of bounded linear operators. Linear Algebra Appl. 506, 551–563 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rao, T.S.S.R.K.: Smooth points in spaces of operators. Linear Algebra Appl. 517, 129–133 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rao, T.S.S.R.K.: On a theorem of Abatzoglou for operators on abstract L and M-spaces. J. Math. Anal. Appl. 453, 1000–1004 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ruess, W.M., Stegall, C.P.: Extreme points in duals of operator spaces. Math. Ann. 261, 535–546 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sain, D.: Birkhoff–James orthogonality of linear operators on finite dimensional Banach spaces. J. Math. Anal. Appl. 447, 860–866 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Singer, I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Grundlehren der mathematischen Wissenschaften, vol. 171. Springer, Berlin (1970)CrossRefzbMATHGoogle Scholar
  18. 18.
    Sain, D., Paul, K., Mal, A.: A complete characterization of Birkhoff–James orthogonality in infinite dimensional normed space. J. Oper. Theory 80(2), 399–413 (2018)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Sain, D., Paul, K., Mal, A., Ray, A.: A complete characterization of smoothness in the space of bounded linear operators. Linear Multilinear Algebra (2019). Google Scholar
  20. 20.
    Wójcik, P.: Birkhoff orthogonality in classical M-ideals. J. Aust. Math. Soc. 103, 279–288 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsJadavpur UniversityKolkataIndia
  2. 2.Theoretical Statistics and Mathematics UnitIndian Statistical InstituteBengaluruIndia
  3. 3.Department of MathematicsIndian Institute of ScienceBengaluruIndia

Personalised recommendations