Numerical index and Daugavet property of operator ideals and tensor products

TL;DR

This paper investigates the numerical index and Daugavet property in operator ideals and tensor products of Banach spaces, establishing bounds and inheritance properties that deepen understanding of their geometric and functional structure.

Contribution

It provides new bounds for the numerical index of operator ideals and tensor products, and characterizes conditions under which the Daugavet property is inherited between factors.

Findings

Numerical index of operator ideals is bounded by those of domain and range.

Numerical index of certain operator ideals is bounded by the dual of the domain.

Daugavet property can be inherited between factors in tensor products under specific conditions.

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 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 Fr\'{e}chet 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 Fr\'{e}chet differentiability of the norm, then the other factor has the Daugavet property.