An elementary approach to the Daugavet equation

TL;DR

This paper provides a simple, necessary and sufficient condition for the Daugavet equation to hold for bounded linear operators on C(S), with applications to weakly compact operators and those factoring through c0, simplifying existing proofs.

Contribution

It introduces an elementary approach to characterize when the Daugavet equation holds, streamlining proofs of several classical results.

Findings

Characterizes the Daugavet equation for bounded linear operators on C(S).

Applies the characterization to weakly compact operators.

Applies the characterization to operators factoring through c0.

Abstract

Let $T\dopu C(S)\to C(S)$ be a bounded linear operator. We present a necessary and sufficient condition for the so-called Daugavet equation $$ \|\Id+T\| = 1+\|T\| $$ to hold, and we apply it to weakly compact operators and to operators factoring through $c_{0}$. Thus we obtain very simple proofs of results by Foias, Singer, Pelczynski, Holub and others.