Abstract

We explore the relation between the orthogonality of bounded linear operators in the space of operators and that of elements in the ground space. To be precise, we study if T,A∈L(X,Y)T, A \in \mathbb{L}(\mathbb{X}, \mathbb{Y}) satisfy T⊥BAT \bot_B A, then whether there exists x∈Xx \in \mathbb{X} such that Tx⊥BAxTx\bot_B Ax with ∥x∥=1\|x\| =1, ∥Tx∥=∥T∥\|Tx\| = \|T\|, where X,Y\mathbb{X}, \mathbb{Y} are normed linear spaces. In this context, we introduce the notion of Property PnP_n for a Banach space and illustrate its connection with orthogonality of a bounded linear operator between Banach spaces. We further study Property PnP_n for various polyhedral Banach spaces.

Author information

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

Kallol Paul

Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India

kalloldada@gmail.com

Debmalya Sain

Department of Mathematics, Indian Institute of Science, Bengaluru 560012, Karnataka, India

saindebmalya@gmail.com

Keywords

  • Orthogonality
  • linear operators
  • norm attainment
  • polyhedral Banach spaces

Mathematics Subject Classification

46B20; 47L05

References

  1. R. Bhatia, P. Šemrl: Orthogonality of matrices and some distance problems, Linear Algebra Appl. 287/1-3 (1999) 77–85.https://doi.org/10.1016/S0024-3795(98)10134-9Google Scholar
  2. 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
  3. C. K. Li, H. Schneider: Orthogonality of matrices, Linear Algebra Appl. 347 (2002) 115–122.https://doi.org/10.1016/S0024-3795(01)00530-4Google Scholar
  4. K. Paul: Translatable radii of an operator in the direction of another operator, Sci. Math. 2/1 (1999) 119–122.Google Scholar
  5. K. Paul, Sk. M. Hossein, K. C. Das: Orthogonality on B(H, H) and minimal-norm Operator, J. Analysis Appl. 6/3 (2008) 169–178.Google Scholar
  6. D. Sain: Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, J. Math. Analysis Appl. 447 (2017) 860–866.https://doi.org/10.1016/j.jmaa.2016.10.064Google Scholar
  7. D. Sain, K. Paul: Operator norm attainment and inner product spaces, Linear Algebra Appl. 439/8 (2013) 2448–2452.https://doi.org/10.1016/j.laa.2013.07.008Google Scholar
  8. D. Sain, K. Paul, P. Bhunia, S. Bag: On the numerical index of polyhedral Banach spaces, Linear Algebra Appl. 577 (2019) 121–133.https://doi.org/10.1016/j.laa.2019.04.024Google Scholar
  9. D. Sain, K. Paul, S. Hait: Operator norm attainment and Birkhoff-James orthogonality, Linear Algebra Appl. 476 (2015) 85–97.https://doi.org/10.1016/j.laa.2015.03.002Google Scholar

Suggested citation

A. Ray, K. Paul, D. Sain, S. Dey. “Some Remarks on Orthogonality of Bounded Linear Operators.” Journal of Convex Analysis 29 (2022), No. 1, 165–181. https://doi.org/10.68381/jca29010

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Published by Heldermann Verlag, 2022. Rights now held by Banach Press.