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.

我们表明，任何算子理想的数值索引均小于或等于域空间和范围空间的数值索引的最小值。此外，我们表明，紧致算子的理想或弱紧致算子的理想的指数小于或等于域空间对偶的指数，并且该结果提供了有趣的例子。我们还表明，Banach空间的射影或射量张量积的数值指标小于或等于任何因素的数值指标。最后，我们证明，如果两个Banach空间的射影张量积具有Daugavet属性，并且其中一个因子的单位球是可切片可数确定的，或者其对偶包含范数的Fréchet可微性，那么另一个因素继承了Daugavet属性。如果两个Banach空间的内射张量积具有Daugavet属性，并且其中一个因素包含范数的Fréchet可微性点，则另一个因素具有Daugavet属性。