An elementary approach to the Daugavet equation

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.