On the convex compactness property for the strong operator topology

Abstract

In the strong operator topology, the space $K(X, Y)$ of compact operators between two Banach spaces X, Y is not complete, not even sequentially complete. It is, however, Mackey complete, i.e., every bounded closed absolutely convex subset is a Banach disk (cf. [4]). In this paper we show that $K(X, Y)$ , with the strong operator topology, has a stronger completeness property, namely the convex compactness property (see the definition below).This property is also true for the space of weakly compact operators ([9]).These considerations concerning the convex compactness property of $K( X, Y)$ and of other subspaces of $L(X, Y)$ (the space of all continuous linear operators) in the strong operator topology were motivated by the paper of Weis [ 11].They originated from the context of the perturbation theory of $C₀$-semigroups, in particular from the application to the neutron transport equation. We refer to [11] as well as to the references quoted there for motivation.