There are two notions of approximate Birkhoff-James orthogonality sets in a Banach space. A geometric study of them has recently been done by D. Sain, K. Paul and A. Mal [On approximate Birkhoff-James orthogonality and normal cones in a normed space, J. Convex Analysis 26 (2019) 341–351]. We here characterize completely one of these sets in terms of normal cones. We obtain a uniqueness criterion between approximate orthogonality sets and normal cones in a Banach space. We also investigate the action of a bounded linear operator on approximate Birkhoff-James orthogonality sets.

Contact details are reproduced from the original publication and may be historical.

Jeet Sen

Dept. of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India

senet.jeet@gmail.com

Arpita Mal

Dept. of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India

arpitamalju@gmail.com

Kallol Paul

Dept. of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India

kalloldada@gmail.com
  1. G. Birkhoff: Orthogonality in linear metric spaces, Duke Math. J. 1 (1935) 169–172.https://doi.org/10.1215/S0012-7094-35-00115-6Google Scholar
  2. J. Chmieliński: On an ϵ-Birkhoff orthogonality, J. Inequal. Pure Appl. Math. 6(3) (2005) Article 79.Google Scholar
  3. J. Chmieliński, T. Stypuła and P. Wójcik: Approximate orthogonality in normed spaces and its applications, Linear Algebra Appl. 531 (2017) 305–317.https://doi.org/10.1016/j.laa.2017.06.001Google Scholar
  4. S. S. Dragomir: On approximation of continuous linear functionals in normed linear spaces, An. Univ. Timisoara Ser. Stiint. Mat. 29(1) (1991) 51–58.Google Scholar
  5. R. C. James: Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947) 265–292.https://doi.org/10.1090/S0002-9947-1947-0021241-4Google Scholar
  6. A. Mal, D. Sain, K. Paul: On some geometric properties of operator spaces, Banach J. Math. Analysis 13(1) (2019) 174–191.https://doi.org/10.1215/17358787-2018-0021Google Scholar
  7. D. Sain: Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, J. Math. Anal. Appl. 447(2) (2017) 860–866.https://doi.org/10.1016/j.jmaa.2016.10.064Google Scholar
  8. D. Sain, P. Ghosh, K. Paul: On symmetry of Birkhoff-James orthogonality of linear operators on finite-dimensional real Banach spaces, Oper. Matrices 11(4) (2017) 1087–1095.https://doi.org/10.7153/oam-2017-11-75Google Scholar
  9. D. Sain, K. Paul, A. Mal: On approximate Birkhoff-James orthogonality and normal cones in a normed space, J. Convex Analysis 26(1) (2019) 341–351.https://doi.org/10.68381/jca26019Google Scholar

J. Sen, A. Mal, K. Paul. “Characterization of Approximate Birkhoff-James Orthogonality Sets in a Banach Space.” Journal of Convex Analysis 27 (2020), No. 3, 881–892. https://doi.org/10.68381/jca27044