Numerical Index and Daugavet Property of Operator Ideals and Tensor Products Martín Suárez, Miguel Meri De La Maza, Francisco Javier Quero de la Rosa, Alicia Banach spaces Daugavet property Numerical index Numerical radius Numerical range Operator ideal Projective and injective tensor Projective and injective tensor products Licely countably determined sets and operators The authors thank Abraham Rueda Zoca for many conversations on the topic of this manuscript. 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 and the range. 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, 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. 2021-03-23T12:18:39Z 2021-03-23T12:18:39Z 2020-05-26 info:eu-repo/semantics/article Publisher version: 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] http://hdl.handle.net/10481/67613 10.1007/s00009-021-01721-9 eng http://creativecommons.org/licenses/by-nc-nd/3.0/es/ info:eu-repo/semantics/openAccess Atribución-NoComercial-SinDerivadas 3.0 España Springer Nature