A note on uniformly dominated sets of summing operators

Abstract

Let Y be a Banach space that has no finite cotype and p a real number satisfying 1≤p<∞. We prove that a set ℳ⊂Πp(X,Y) is uniformly dominated if and only if there exists a constant C>0 such that, for every finite set {(xi,Ti):i=1,…,n}⊂X×ℳ, there is an operator T∈Πp(X,Y) satisfying πp(T)≤C and ‖Tixi‖≤‖Txi‖ for i=1,…,n.