SpringerLink
Log in

Birkhoff-James orthogonality and its local symmetry in some sequence spaces

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We study Birkhoff-James orthogonality and its local symmetry in some sequence spaces namely \(\ell _p,\) for \(1\le p\le \infty \), \(p\ne 2\), c, \(c_0\) and \(c_{00}\). Using the characterization of the local symmetry of Birkhoff-James orthogonality, we characterize isometries of each of these spaces onto itself and obtain the Banach-Lamperti theorem for onto operators on the sequence spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arambašić, L., Rajić, R.: On symmetry of the (strong) Birkhoff-James orthogonality in Hilbert \(C^*\)-modules. Ann. Funct. Anal. 7(1), 17–23 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Chattopadhyay, A., Sain, D., Senapati, T.: Characterization of symmetric points in \( l_p^n \)-spaces. Linear Multilinear Algebra 69(16), 2998–3009 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chmieliński, J., Wójcik, P.: On a \(\rho \)-orthogonality. Aequationes Math. 80(1–2), 45–55 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chmieliński, J., Wójcik, P.: \(\rho \)-orthogonality and its preservation-revisited. Banach Center Publ. 1(99), 17–30 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Comfort, W.W.: Ultrafilters : some old and some new results. Bull. Amer. Math. Soc. 83(4), 417–455 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Comfort, W.W., Negrepontis, S.: The theory of ultrafilters, vol. 211. Springer Science & Business Media, New York (2012)

    MATH  Google Scholar 

  8. Ghosh, P., Sain, D., Paul, K.: On symmetry of Birkhoff-James orthogonality of linear operators. Adv. Oper. Theory 2, 428–434 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Ghosh, P., Paul, K., Sain, D.: Symmetric properties of orthogonality of linear operators on \((\mathbb{R} ^ n,\Vert.\Vert _1)\). Novi Sad J. Math. 47(2), 41–46 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. James, R.C.: Inner product in normed linear spaces. Bull. Amer. Math. Soc. 53(6), 559–566 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  11. James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Amer. Math. Soc. 61(2), 265–292 (1947)

    Article  MathSciNet  Google Scholar 

  12. Komuro, N., Saito, K.-S., Tanaka, R.: Left symmetric points for Birkhoff orthogonality in the preduals of von Neumann algebras. Bull. Aust. Math. Soc. 98(3), 494–501 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Komuro, N., Saito, K.-S., Tanaka, R.: Symmetric points for (strong) Birkhoff orthogonality in von Neumann algebras with applications to preserver problems. J. Math. Anal. Appl. 463(2), 1109–1131 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Komuro, N., Saito, K.-S., Tanaka, R.: On symmetry of Birkhoff orthogonality in the positive cones of \(C^*\)-algebras with applications. J. Math. Anal. Appl. 474(2), 1488–1497 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lamperti, J.: On the isometries of certain function-spaces. Pacific J. Math. 8(3), 459–466 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  16. Paul, K., Mal, A., Wójcik, P.: Symmetry of Birkhoff-James orthogonality of operators defined between infinite dimensional Banach spaces. Linear Algebra Appl. 563, 142–153 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sain, D.: Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces. J. Math. Anal. Appl. 447(2), 860–866 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sain, D.: On the norm attainment set of a bounded linear operator. J. Math. Anal. Appl. 457(1), 67–76 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sain, D., Ghosh, P., Paul, K.: On symmetry of Birkhoff-James orthogonality of linear operators on finite-dimensional real Banach spaces. Oper. Matrices 11, 1087–1095 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sain, D., Paul, K., Mal, A., Ray, A.: A complete characterization of smoothness in the space of bounded linear operators. Linear Multilinear Algebra 68(12), 2484–2494 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sain, D., Roy, S., Bagchi, S., Balestro, V.: A study of symmetric points in Banach spaces. Linear Multilinear Algebra 70(5), 888–898 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sain, D.: Orthogonality and smoothness induced by the norm derivatives. Rev. R. Acad. Cienc. Exactas FÍs. Nat. Ser. A Mat RACSAM 115(3), 120 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  23. Turnŝek, A.: A remark on orthogonality and symmetry of operators in B(H). Linear Algebra Appl. 535, 141–150 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Turnsek, A.: On operators preserving James’ orthogonality. Linear Algebra Appl. 407, 189–195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wójcik, P.: Linear map**s preserving \(\rho \)-orthogonality. J. Math. Anal. Appl. 386(1), 171–176 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referee(s) for careful reading and constructive suggestions towards the improvement of the article.

Author information

Author notes

  1. Saikat Roy and Debmalya Sain have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Science, C V Raman Road, Bangalore, Karnataka, 560012, India

    Babhrubahan Bose & Debmalya Sain

  2. Department of Mathematics, Indian Institute of Technology, Powai, Mumbai, Maharashtra, 400076, India

    Saikat Roy

  3. Department of Mathematics, Indian Institute of Information Technology, Yermarsur Campus Road, Raichur, Karnataka, 584135, India

    Babhrubahan Bose, Saikat Roy & Debmalya Sain

Authors
  1. Babhrubahan Bose
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saikat Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Debmalya Sain
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Babhrubahan Bose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bose, B., Roy, S. & Sain, D. Birkhoff-James orthogonality and its local symmetry in some sequence spaces. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 93 (2023). https://doi.org/10.1007/s13398-023-01420-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-023-01420-y

Keywords

Mathematics Subject Classification

Search

Navigation