Narrow operators and rich subspaces of Banach spaces with the Daugavet property

Let $X$ be a Banach space. We introduce a formal approach which seems to be useful in the study of those properties of operators on $X$ which depend only on the norms of images of elements. This approach is applied to the Daugavet equation for norms of operators; in particular we develop a general theory of narrow operators and rich subspaces of $X$ previously studied in the context of the classical spaces $C(K)$ and $L_1(\mu)$.