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Operator Theory, Functional Analysis and Applications pp 293–315Cite as

Birkhäuser

Birkhoff–James Orthogonality and Applications: A Survey

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Part of the book series: Operator Theory: Advances and Applications ((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.

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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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  1. Department of Mathematics, Shiv Nadar University, Dadri, Uttar Pradesh, India

    Priyanka Grover & Sushil Singla

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  1. Priyanka Grover
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Correspondence to Priyanka Grover .

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Editors and Affiliations

  1. Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    M. Amélia Bastos

  2. Departamento de Matemática, Universidade de Aveiro, Aveiro, Portugal

    Luís Castro

  3. Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Lisboa, Portugal

    Alexei Yu. Karlovich

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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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