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Numerical Index and Daugavet Property of Operator Ideals and Tensor Products

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Abstract

We show that the numerical index of any operator ideal is less than or equal to the minimum of the numerical indices of the domain space and the range space. Further, we show that the numerical index of the ideal of compact operators or the ideal of weakly compact operators is less than or equal to the numerical index of the dual of the domain space, and this result provides interesting examples. We also show that the numerical index of a projective or injective tensor product of Banach spaces is less than or equal to the numerical index of any of the factors. Finally, we show that if a projective tensor product of two Banach spaces has the Daugavet property and the unit ball of one of the factor is slicely countably determined or its dual contains a point of Fréchet differentiability of the norm, then the other factor inherits the Daugavet property. If an injective tensor product of two Banach spaces has the Daugavet property and one of the factors contains a point of Fréchet differentiability of the norm, then the other factor has the Daugavet property.

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Acknowledgements

The authors thank Abraham Rueda Zoca for many conversations on the topic of this manuscript. We also thank the referee for the careful reading of the manuscript and for some stylistic remarks.

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Authors and Affiliations

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

    Miguel Martín, Javier Merí & Alicia Quero

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  1. Miguel Martín
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  2. Javier Merí
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  3. Alicia Quero
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Correspondence to Miguel Martín.

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Research partially supported by projects PGC2018-093794-B-I00 (MCIU/AEI/FEDER, UE) and FQM-185 (Junta de Andalucía/FEDER, UE). The third author is also supported by the Ph.D. scholarship FPU18/03057 (MECD)

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Martín, M., Merí, J. & Quero, A. Numerical Index and Daugavet Property of Operator Ideals and Tensor Products. Mediterr. J. Math. 18, 72 (2021). https://doi.org/10.1007/s00009-021-01721-9

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  • DOI: https://doi.org/10.1007/s00009-021-01721-9

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