The Daugavet equation for bounded vector valued functions

TL;DR

This paper investigates the conditions under which the Daugavet and alternative Daugavet equations hold for pairs of nonlinear maps between Banach spaces, providing geometric characterizations and local versions of these properties.

Contribution

It introduces geometric descriptions using nonlinear slices and develops tests for verifying the Daugavet properties in various cases.

Findings

Characterization of Daugavet equations via nonlinear slices

Introduction of local versions of Daugavet properties

Development of practical tests for checking conditions

Abstract

Requirements under which the Daugavet equation and the alternative Daugavet equation hold for pairs of nonlinear maps between Banach spaces are analysed. A geometric description is given in terms of nonlinear slices. Some local versions of these properties are also introduced and studied, as well as tests for checking if the required conditions are satisfied in relevant cases.