Spaces of compact operators and their dual spaces

Abstract

Theω′-topology on the spaceL(X, Y) of bounded linear operators from the Banach spaceX into the Banach spaceY is discussed in [10]. Let ℒ^w' (X, Y) denote the space of allT∈L(X, Y) for which there exists a sequence of compact linear operators (T _n)⊂K(X, Y) such thatT=ω′−lim_nT_n and let$|||T|||: = \{ \sup _n ||T_n ||:T_n \in K(X,Y),T_n \mathop \to \limits^{w'} T\}$. We show that$(\mathcal{L}^{w'} ,||| \cdot |||)$ is a Banach ideal of operators and that the continuous dual spaceK(X, Y)* is complemented in$(\mathcal{L}^{w'} (X,Y),||| \cdot |||)*$. This results in necessary and sufficient conditions forK(X, Y) to be reflexive, whereby the spacesX andY need not satisfy the approximation property. Similar results follow whenX andY are locally convex spaces.