Uniformly Summing Sets of Operators on Spaces of Continuous Functions

Abstract

Let X and Y be Banach spaces. A set ℳ of 1-summing operators from X into Y is said to be uniformly summing if the following holds: given a weakly 1-summing sequence (xn) in X, the series ∑n‖Txn‖ is uniformly convergent in T∈ℳ. We study some general properties and obtain a characterization of these sets when ℳ is a set of operators defined on spaces of continuous functions.