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


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.

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  1. Birkhoff, G.: Orthogonality in linear metric spaces. Duke Math. J. 1, 169–172 (1935)

    Article  MathSciNet  Google Scholar 

  2. Chmieliński, J.: On an \(\epsilon \)-Birkhoff orthogonality. J. Inequal. Pure Appl. Math. 6(3), Article 79 (2005)

  3. Chmieliński, J., Stypuła, T., Wójcik, P.: Approximate orthogonality in normed spaces and its applications. Linear Algebra Appl. 531, 305–317 (2017)

    Article  MathSciNet  Google Scholar 

  4. Dragomir, S.S.: On approximation of continuous linear functionals in normed linear spaces. An. Univ. Timisoara Ser. Stiint. Mat. 29, 51–58 (1991)

    MathSciNet  MATH  Google Scholar 

  5. Grza̧ślewicz, R., Younis, R.: Smooth points and M-ideals. J. Math. Anal. Appl. 175, 91–95 (1993)

    Article  MathSciNet  Google Scholar 

  6. Holmes, R.B.: Geometric Functional Analysis and Its Applications. Graduate Texts in Mathematics, vol. 24. Springer, New York (1975). x+246pp

    Book  Google Scholar 

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

    Book  Google Scholar 

  8. James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 61, 265–292 (1947)

    Article  MathSciNet  Google Scholar 

  9. Martin, M.: Norm-attaining compact operators. J. Funct. Anal. 267, 1585–1592 (2014)

    Article  MathSciNet  Google Scholar 

  10. Mal, A., Sain, D., Paul, K.: On some geometric properties of operator spaces. Banach J. Math. Anal. 13(1), 174–191 (2019)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  12. Paul, K., Sain, D., Ghosh, P.: Birkhoff–James orthogonality and smoothness of bounded linear operators. Linear Algebra Appl. 506, 551–563 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Sain, D.: Birkhoff–James orthogonality of linear operators on finite dimensional Banach spaces. J. Math. Anal. Appl. 447, 860–866 (2017)

    Article  MathSciNet  Google Scholar 

  17. Singer, I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Grundlehren der mathematischen Wissenschaften, vol. 171. Springer, Berlin (1970)

    Book  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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).

    Article  Google Scholar 

  20. Wójcik, P.: Birkhoff orthogonality in classical M-ideals. J. Aust. Math. Soc. 103, 279–288 (2017)

    Article  MathSciNet  Google Scholar 

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Correspondence to Kallol Paul.

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Communicated by G. Teschl.

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The first and second author acknowledges the generosity of Indian Statistical Institute, Bangalore, and in particular, Professor T. S. S. R. K. Rao, for supporting the visit to the Institute during June 2018. This research paper originated from that visit. First author would like to thank UGC, Govt. of India for the financial support. The research of Prof. Paul is supported by Project MATRICS (MTR/2017/000059) of DST, Govt. of India. The research of Dr. Debmalya Sain is sponsored by Dr. D. S. Kothari Postdoctoral Fellowship under the mentorship of Prof. Gadadhar Misra. Dr. Sain feels elated to acknowledge the motivating presence of his younger brother Debdoot in every sphere of his life.

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Mal, A., Paul, K., Rao, T.S.S.R.K. et al. Approximate Birkhoff–James orthogonality and smoothness in the space of bounded linear operators. Monatsh Math 190, 549–558 (2019).

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  • Orthogonality
  • Linear operators
  • M-ideal
  • L-ideal
  • Smoothness

Mathematics Subject Classification

  • Primary 46B28
  • Secondary 47L05
  • 46B20