A note on p-limited sets

Given p ⩾ 1 , a subset A of a Banach space X is said to be p-limited if for every weakly p-summable sequence ( x n ⁎ ) in X ⁎ there exists ( α n ) ∈ ℓ p such that | 〈 x n ⁎ , x 〉 | ⩽ α n for all x ∈ A and n ∈ N . It is showed that p-limited sets are q-limited whenever p < q and Banach spaces enjoying the property that every q-limited subset is p-limited are characterized. We also prove that an operator has p-summing adjoint if and only if it maps relatively compact sets to p-limited sets.