Numerical Index and Daugavet Property of Operator Ideals and Tensor Products

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