A Note on Uniformly Dominated Sets of Summing Operators

for allx ∈X and all T ∈ . Since the appearance of Grothendieck-Pietsch’s domination theorem for p-summing operators, there is a great interest in finding out the structure of uniformly dominated sets. We will denote by p(μ) the set of all operators T ∈ Πp(X,Y) satisfying (1.1) for all x ∈ X. It is easy to prove that p(μ) is absolutely convex, closed, and bounded (for the p-summing norm). In [4], the authors consider the case p = 1 and prove that ⊂ Πp(X,Y) is uniformly dominated if and only if ⋃ T∈ T∗(BY∗) lies in the range of a vector measure of bounded variation and valued in X∗. In [3], the following sufficient condition is proved: “let ⊂ Πp(X,Y) and 1 ≤ p < ∞. Suppose that there is a positive constant C > 0 such that, for every finite set {x1, . . . ,xn} of X, there exists Q∈ satisfying πp(Q)≤ C and