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Orthogonality and norm attainment of operators in semi-Hilbertian spaces

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Abstract

We study the semi-Hilbertian structure induced by a positive operator A on a Hilbert space \({\mathbb {H}}.\) Restricting our attention to \(A-\)bounded positive operators, we characterize the norm attainment set and also investigate the corresponding compactness property. We obtain a complete characterization of the \(A-\)Birkhoff–James orthogonality of \(A-\)bounded operators under an additional boundedness condition. This extends the finite-dimensional Bhatia-\( \breve{S} \)emrl Theorem verbatim to the infinite-dimensional setting.

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Acknowledgements

The research of Jeet Sen is supported by CSIR, Govt. of India. The research of Prof. Kallol Paul is supported by project MATRICS (MTR/2017/000059) of SERB, DST, Govt. of India.

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Authors and Affiliations

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

    Jeet Sen

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

    Debmalya Sain & Kallol Paul

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  1. Jeet Sen
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  2. Debmalya Sain
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  3. Kallol Paul
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Correspondence to Kallol Paul.

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Communicated by Jacek Chmielinski.

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Sen, J., Sain, D. & Paul, K. Orthogonality and norm attainment of operators in semi-Hilbertian spaces. Ann. Funct. Anal. 12, 17 (2021). https://doi.org/10.1007/s43034-020-00104-7

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  • DOI: https://doi.org/10.1007/s43034-020-00104-7

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