On the extension problem between separable smooth Banach spaces with RNP

Abstract

In this paper, we study the extension of isometric operators between unit spheres of separable smooth Banach spaces with the Radon-Nikodym property (RNP). We show that if there is a surjective isometric operator between unit spheres of separable smooth Banach spaces with RNP, then there exists a codimension one subspace which is linearly isometric to another space. We also show that the surjective isometric operator between unit spheres of Banach spaces with dense smooth points can be extended isometrically to the whole space under some condition.