The Daugavet equation in uniformly convex Banach spaces

Abstract

It is shown that a continuous operator T: X → X on a uniformly convex Banach space satisfies the Daugavet equation ∥ I + T∥ = 1 + ∥ T∥ if and only if the norm ∥ T∥ of the operator lies in the spectrum of T. Specializing this result to compact operators, we see that a compact operator on a uniformly convex Banach space satisfies the Daugavet equation if and only if its norm is an eigenvalue. The latter conclusion is in sharp contrast with the standard facts on the Daugavet equation for the spaces L 1( μ) and L ∞( μ). A discussion of the Daugavet property in the latter spaces is also included in the paper.