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On the Compact Operators Case of the Bishop–Phelps–Bollobás Property for Numerical Radius

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Abstract

We study the Bishop–Phelps–Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that \(C_0(L)\) spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space L. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of \(C_0(L)\) spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of \(\ell _1\) have the BPBp-nu for compact operators.

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Acknowledgements

The authors would like to thank Bill Johnson for kindly answering several inquiries.

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100, Burjasot, Valencia, Spain

    Domingo García, Manuel Maestre & Óscar Roldán

  2. Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain

    Miguel Martín

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  1. Domingo García
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  2. Manuel Maestre
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  3. Miguel Martín
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  4. Óscar Roldán
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Correspondence to Domingo García.

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The first and second authors were supported by MINECO and FEDER Project MTM2017-83262-C2-1-P and by Prometeo PROMETEO/2017/102. The third author was supported by Projects PGC2018-093794-B-I00 (MCIU/AEI/FEDER, UE), A-FQM-484-UGR18 (Universidad de Granada and Junta de Analucía/FEDER, UE), and FQM-185 (Junta de Andalucía/FEDER, UE). The fourth author was supported by the Spanish Ministerio de Ciencia, Innovación y Universidades, Grant FPU17/02023, and by MINECO and FEDER Project MTM2017-83262-C2-1-P.

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García, D., Maestre, M., Martín, M. et al. On the Compact Operators Case of the Bishop–Phelps–Bollobás Property for Numerical Radius. Results Math 76, 122 (2021). https://doi.org/10.1007/s00025-021-01430-5

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