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Birkhoff–James Orthogonality and Applications: A Survey

Part of the Operator Theory: Advances and Applications book series (OT,volume 282)

Abstract

In the last few decades, the concept of Birkhoff–James orthogonality has been used in several applications. In this survey article, the results known on the necessary and sufficient conditions for Birkhoff–James orthogonality in certain Banach spaces are mentioned. Their applications in studying the geometry of normed spaces are given. The connections between this concept of orthogonality, and the Gateaux derivative and the subdifferential set of the norm function are provided. Several interesting distance formulas can be obtained using the characterizations of Birkhoff–James orthogonality, which are also mentioned. In the end, some new results are obtained.

Keywords

  • Orthogonality
  • Tangent hyperplane
  • Smooth point
  • Faces of unit ball
  • Gateaux differentiability
  • Subdifferential set
  • State on a C -algebra
  • Cyclic representation
  • Norm-parallelism
  • Conditional expectation

Mathematics Subject Classification (2010)

  • Primary 15A60
  • 41A50
  • 46B20
  • 46L05
  • 46L08; Secondary 46G05
  • 47B47

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Notes

  1. 1.

    We learnt this characterization of orthogonality in \({\mathbb {R}}^n\) from Amber Habib.

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Acknowledgements

The authors would like to thank Amber Habib and Ved Prakash Gupta for many useful discussions. The authors would also like to acknowledge very helpful comments by the referees.

The research of P. Grover is supported by INSPIRE Faculty Award IFA14-MA-52 of DST, India, and by Early Career Research Award ECR/2018/001784 of SERB, India.

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Grover, P., Singla, S. (2021). Birkhoff–James Orthogonality and Applications: A Survey. In: Bastos, M.A., Castro, L., Karlovich, A.Y. (eds) Operator Theory, Functional Analysis and Applications. Operator Theory: Advances and Applications, vol 282. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-51945-2_15

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