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Norm derivatives and geometry of bilinear operators

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Abstract

We study the norm derivatives in the context of Birkhoff–James orthogonality in real Banach spaces. As an application of this, we obtain a complete characterization of the left-symmetric points and the right-symmetric points in a real Banach space in terms of the norm derivatives. We obtain a complete characterization of strong Birkhoff–James orthogonality in \(\ell _1^n\) and \(\ell _\infty ^n\) spaces. We also obtain a complete characterization of the orthogonality relation defined by the norm derivatives in terms of some newly introduced variation of Birkhoff–James orthogonality. We further study Birkhoff–James orthogonality, approximate Birkhoff–James orthogonality, smoothness and norm attainment of bounded bilinear operators between Banach spaces.

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Acknowledgements

Dr. Sain feels elated to acknowledge his childhood friend Tamal Bandyopadhyay for his inspiring presence in every sphere of his life. The research of Dr. Divya Khurana and Dr. Debmalya Sain is sponsored by Dr. D. S. Kothari Postdoctoral Fellowship under the mentorship of Professor Gadadhar Misra.

Funding

Funding was provided by University Grants Commission.

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Authors and Affiliations

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

    Divya Khurana & Debmalya Sain

Authors
  1. Divya Khurana
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  2. Debmalya Sain
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Corresponding author

Correspondence to Divya Khurana.

Additional information

Communicated by Pedro Tradacete.

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Khurana, D., Sain, D. Norm derivatives and geometry of bilinear operators. Ann. Funct. Anal. 12, 49 (2021). https://doi.org/10.1007/s43034-021-00134-9

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  • DOI: https://doi.org/10.1007/s43034-021-00134-9

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