© Hindawi Publishing Corp. UNIFORMLY SUMMING SETS OF OPERATORS ON SPACES OF CONTINUOUS FUNCTIONS

Let X and Y be Banach spaces. A set 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 ∈. We study some general properties and obtain a characterization of these sets when is a set of operators defined on spaces of continuous functions. 2000 Mathematics Subject Classification: 47B38, 47B10. 1. Introduction. Throughout this paper, X and Y will be Banach spaces. If X is a Banach space, BX = {x ∈X: ‖x ‖ ≤ 1} will denote its closed unit ball and X ∗ will be the topological dual of X. Given a real number p ∈ [1,∞), a (linear) operator T: X → Y is said to be p-summing if there exists a constant C> 0 such that ( n