Unconditionally Convergent Series of Operators on Banach Spaces

We show that any series nKn of operators in L X,Y.that is unconditionally convergent in the weak operator topology and satisfies the condition that ngF Kn is a compact operator for every index set F:N is unconditionally convergent in the uniform operator topology if and only if X*, the dual space of the Banach space X, contains no copy of c0.