Narrow operators and rich subspaces of Banach spaces with the Daugavet property (Q2759179)

The authors introduce concepts of narrow operators and rich subspaces of Banach spaces which are shown to be useful for studying properties which only depend on the norms of the images of the elements. In particular, the Daugavet equation \(\|I+T\|= 1+\|T\|\) is studied in this context. It is shown that narrow operators on a space \(X\) satisfy the Daugavet equation. Strong Radon-Nikodým operators and operators not fixing a copy of \(\ell_1\) are shown to be narrow. The Daugavet equation passes from spaces to rich subspaces.