A generalization of a Daugavet theorem with applications to the space \(C\) geometry (Q1128371)

Let \(Y\) be a Banach space, \(X\) be its subspace, \(L(X,Y)\) be the space of bounded operators from \(X\) into \(Y\), and let \(J \in L(X,Y)\) be the natural embedding. A pair \((X,Y)\) is said to be a Daugavet pair (\((X,Y) \in D\)) if for every compact operator \(T \in L(X,Y)\) one has \(\| J+T\| =1+\| T\| \). A space \(X\) has the Daugavet property if \((X,X) \in D\). Main results: Theorem 1. Suppose that \(X=C\), \(Y\) is separable, and \(Y \supset X\). Then there exists an equivalent norm on \(Y\) for which \((X,Y) \in D\). Theorem 2. If \((X,Y) \in D\), then \(Y\) has no unconditional basis. The author notes that using the results of the article it is easy to show that the space \(C\) is not only nonembeddible in a space with unconditional basis but also cannot serve as a subspace of an unconditional direct sum of reflexive spaces.