The Daugavet equation in Banach spaces with alternatively convex-smooth duals

Abstract

This short paper gives a necessary and sufficient condition for the Daugavet equation $\Vert I+T\Vert =1+\Vert T\Vert$. A new characterization of the solution of the Daugavet equation in terms of invariant affine subspaces is given. We also study the notions of alternatively convex or smooth (acs) and locally uniformly alternatively convex or smooth (luacs).