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Norm-Attaining Tensors and Nuclear Operators

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Abstract

Given two Banach spaces X and Y, we introduce and study a concept of norm-attainment in the space of nuclear operators \(\mathcal N(X,Y)\) and in the projective tensor product space \(X\widehat{\otimes }_\pi Y\). We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in \(\mathcal N(X,Y)\) and in \(X\widehat{\otimes }_\pi Y\) is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces X and Y which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces X and Y failing the approximation property in such a way that the class of elements in \(X\widehat{\otimes }_\pi Y\) which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of norm-attaining operators throughout the paper.

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Notes

  1. The authors are thankful to the referee who provided us this example.

  2. It is worth mentioning that this question was posed by the authors in a preliminary version of this manuscript; they thank the anonymous referee who answered it negatively.

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Acknowledgements

The authors are grateful to William B. Johnson for pointing our an error in the proof of their original version of Lemma 5.3. They are also grateful to Gilles Godefroy, Petr Hájek, and Tommaso Russo for giving proper references for the metric \(\pi \)-property. They also would like to thank Richard Aron, Ginés López Pérez, and Miguel Martín for fruitful conversations on the topic of the paper. Finally, the authors want to thank the anonymus referee for his/her valuable mathematical contributions and the linguistic suggestions which have highly improved the quality and the exposition of the paper.

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Authors and Affiliations

  1. Departament de Matemàtiques and Institut Universitari de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Campus del Riu Sec. s/n, 12071, Castelló, Spain

    Sheldon Dantas

  2. Basic Science Research Institute and Department of Mathematics, POSTECH, Pohang, 790-784, Republic of Korea

    Mingu Jung

  3. Departamento de Análisis Matemático, Universitat de València, Valencia, Spain

    Óscar Roldán

  4. Universidad de Murcia, Departamento de Matemáticas, Campus de Espinardo, 30100, Murcia, Spain

    Abraham Rueda Zoca

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  1. Sheldon Dantas
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  2. Mingu Jung
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  3. Óscar Roldán
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  4. Abraham Rueda Zoca
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Correspondence to Mingu Jung.

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The first author was supported by Spanish AEI Project PID2019 - 106529GB - I00 / AEI / 10.13039/501100011033, by the project OPVVV CAAS CZ.02.1.01/0. 0/0.0/16_019/0000778 and by the Estonian Research Council grant PRG877. The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A2C1003857). The third author was supported by the Spanish Ministerio de Universidades, grant FPU17/02023, and by project MTM2017-83262-C2-1-P/MCIN/AEI/10.13039/501100011033 (FEDER). The fourth author was supported by Juan de la Cierva-Formación fellowship FJC2019-039973/ AEI / 10.13039/501100011033, by MTM2017-86182-P (Government of Spain, AEI/FEDER, EU), by MICINN (Spain) Grant PGC2018-093794-B-I00 (MCIU, AEI, FEDER, UE), by Fundación Séneca, ACyT Región de Murcia grant 20797/PI/18, by Junta de Andalucía Grant A-FQM-484-UGR18 and by Junta de Andalucía Grant FQM-0185.

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Dantas, S., Jung, M., Roldán, Ó. et al. Norm-Attaining Tensors and Nuclear Operators. Mediterr. J. Math. 19, 38 (2022). https://doi.org/10.1007/s00009-021-01949-5

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