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Approximate Birkhoff–James orthogonality and smoothness in the space of bounded linear operators

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Abstract

We study approximate Birkhoff–James orthogonality of bounded linear operators defined between normed linear spaces \(\mathbb {X}\) and \(\mathbb {Y}.\) As an application of the results obtained, we characterize smoothness of a bounded linear operator T under the condition that \(\mathbb {K}(\mathbb {X},\mathbb {Y}),\) the space of compact linear operators is an M-ideal in \(\mathbb {L}(\mathbb {X},\mathbb {Y}),\) the space of bounded linear operators.

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Authors and Affiliations

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

    Arpita Mal & Kallol Paul

  2. Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bengaluru, India

    T. S. S. R. K. Rao

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

    Debmalya Sain

Authors
  1. Arpita Mal
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  2. Kallol Paul
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  3. T. S. S. R. K. Rao
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  4. Debmalya Sain
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Correspondence to Kallol Paul.

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Communicated by G. Teschl.

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The first and second author acknowledges the generosity of Indian Statistical Institute, Bangalore, and in particular, Professor T. S. S. R. K. Rao, for supporting the visit to the Institute during June 2018. This research paper originated from that visit. First author would like to thank UGC, Govt. of India for the financial support. The research of Prof. Paul is supported by Project MATRICS (MTR/2017/000059) of DST, Govt. of India. The research of Dr. Debmalya Sain is sponsored by Dr. D. S. Kothari Postdoctoral Fellowship under the mentorship of Prof. Gadadhar Misra. Dr. Sain feels elated to acknowledge the motivating presence of his younger brother Debdoot in every sphere of his life.

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Mal, A., Paul, K., Rao, T.S.S.R.K. et al. Approximate Birkhoff–James orthogonality and smoothness in the space of bounded linear operators. Monatsh Math 190, 549–558 (2019). https://doi.org/10.1007/s00605-019-01289-3

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  • DOI: https://doi.org/10.1007/s00605-019-01289-3

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