Duality and Geometry of Spaces of Compact Operators

Abstract

This chapter discusses the problem of how—for given Banach spaces X and Y—the space K(X,Y) of compact linear operators from X into Y and its dual space reflect the geometric and topological properties of X and Y and their respective duals. Because x and Y are closed linear subspaces of K(X,Y), they inherit trivially properties such as non-containment of t 1 , the dual having the Radon-Nikodym property, weak sequential completeness, or reflexi­ vity, from K(X,Y). The natural question thus is to find out which additional conditions are needed for the reverse implications: how can geometric and topological properties of K(X,Y) and its dual be re­ covered from the corresponding properties of (the presumably well known spaces) X and Y and their duals? The chapter answers these queations.