Slice continuity for operators and the Daugavet property for bilinear maps

TL;DR

This paper introduces the concept of slice continuity for operators in Banach spaces with the Daugavet property, demonstrating its implications for weakly compact operators and bilinear maps, and exploring $p$-convexifications.

Contribution

It defines slice continuity in the context of the Daugavet property and applies it to characterize the property for bilinear maps and certain Banach function spaces.

Findings

Daugavet equation holds for weakly compact operators under slice continuity.

Characterization of the Daugavet property for bilinear maps.

Description of $p$-convexifications of the Daugavet equation.

Abstract

We introduce and analyse the notion of slice continuity between operators on Banach spaces in the setting of the Daugavet property. It is shown that under the slice continuity assumption the Daugavet equation holds for weakly compact operators. As an application we define and characterise the Daugavet property for bilinear maps, and we prove that this allows us to describe some $p$-convexifications of the Daugavet equation for operators on Banach function spaces that have recently been introduced.