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Non-rough Norms and Dentability in Spaces of Operators

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Abstract

In this work, we study non-rough norms in L(X, Y),  the space of bounded linear operators between Banach spaces X and Y. We prove that L(X, Y) has non-rough norm if and only if \(X^*\) and Y have non-rough norm. We show that the injective tensor product \(X{\hat{\otimes }}_{\varepsilon } Y\) has non-rough norm if and only if both X and Y have non-rough norm. We also give an example to show that non-rough norms are not stable under projective tensor product. We also study a related concept namely the small diameter properties in the context of \(L(X,Y)^*\). These results lead to a discussion on stability of the small diameter properties for projective and injective tensor product spaces.

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Acknowledgements

We are grateful to the referee for going through our work meticulously and making significant comments. The first author’s research is supported by the dean’s supplemental fund provided by the College of Arts and Sciences, Loyola University, Maryland, USA. The third author’s research is funded by the National Board for Higher Mathematics (NBHM), Department of Atomic Energy (DAE), Government of India, Ref No. 0203/11/2019-R &D-II/9249.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Loyola University, Baltimore, MD, 21210, USA

    Sudeshna Basu

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

    Julio Becerra Guerrero & Juan Miguel Villegas Yeguas

  3. Department of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur Math, Howrah, West Bengal, 711202, India

    Susmita Seal

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  1. Sudeshna Basu
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  2. Julio Becerra Guerrero
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  3. Susmita Seal
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  4. Juan Miguel Villegas Yeguas
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Contributions

All the four authors SS, SB, JBG and JMVY worked together in preparing Sect. 2. Section 3 was contributed by SB and SS. First draft of this article was written by SS and then it was edited and reviewed by SB, JBG and JMVY.

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Correspondence to Susmita Seal.

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Basu, S., Guerrero, J.B., Seal, S. et al. Non-rough Norms and Dentability in Spaces of Operators. Mediterr. J. Math. 20, 338 (2023). https://doi.org/10.1007/s00009-023-02519-7

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  • DOI: https://doi.org/10.1007/s00009-023-02519-7

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