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On a Set of Norm Attaining Operators and the Strong Birkhoff–James Orthogonality

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Abstract

Continuing the study of recent results on the Birkhoff–James orthogonality and the norm attainment of operators, we introduce a property namely the adjusted Bhatia–Šemrl property for operators which is weaker than the Bhatia–Šemrl property. The set of operators with the adjusted Bhatia–Šemrl property is contained in the set of norm attaining ones as it was in the case of the Bhatia–Šemrl property. It is known that the set of operators with the Bhatia–Šemrl property is norm-dense if the domain space X of the operators has the Radon–Nikodým property like finite dimensional spaces, but it is not norm-dense for some classical spaces such as \(c_0\), \(L_1[0,1]\) and C[0, 1]. In contrast with the Bhatia–Šemrl property, we show that the set of operators with the adjusted Bhatia–Šemrl property is norm-dense when the domain space is \(c_0\) or \(L_1[0,1]\). Moreover, we show that the set of functionals having the adjusted Bhatia–Šemrl property on C[0, 1] is not norm-dense but such a set is weak-\(*\)-dense in \(C(K)^*\) for any compact Hausdorff K.

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Funding

The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2022R1A6A3A01086079) and by the Ministry of Education, Science and Technology [NRF-2020R1A2C1A01010377]. The second author was supported by NRF (NRF-2019R1A2C1003857), by POSTECH Basic Science Research Institute Grant (NRF-2021R1A6A1A10042944) and by a KIAS Individual Grant (MG086601) at Korea Institute for Advanced Study. The third author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [NRF-2020R1C1C1A01012267].

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Authors and Affiliations

  1. Department of Mathematics Education, Dongguk University, Seoul, 04620, Republic of Korea

    Geunsu Choi

  2. School of Mathematics, Korea Institute for Advanced Study, Seoul, 02455, Republic of Korea

    Mingu Jung

  3. Department of Mathematics, Chungbuk National University, Cheongju, Chungbuk,  28644, Republic of Korea

    Sun Kwang Kim

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  1. Geunsu Choi
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  2. Mingu Jung
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All authors contributed to the whole part of works together such as the study conception, design and writing. All authors read and approved the final manuscript.

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Correspondence to Sun Kwang Kim.

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Choi, G., Jung, M. & Kim, S.K. On a Set of Norm Attaining Operators and the Strong Birkhoff–James Orthogonality. Results Math 78, 77 (2023). https://doi.org/10.1007/s00025-023-01852-3

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