Unconditionally Convergent Series of Operators on Banach Spaces

Abstract

We show that any series ∑nKnof operators inL(X,Y) that is unconditionally convergent in the weak operator topology and satisfies the condition that ∑n∈FKnis a compact operator for every index setF⊆N is unconditionally convergent in the uniform operator topology if and only ifX*, the dual space of the Banach spaceX, contains no copy ofc0.