On p-Compact Sets in Classical Banach Spaces

Abstract

Given p ≥ 1, we denote by Cp the class of all Banach spaces X satisfying the equality Kp(Y,X) = Πdp(Y,X) for every Banach space Y , Kp (respectively, Πdp ) being the operator ideal of p-compact operators (respectively, of operators with p-summing adjoint). If X belongs to Cp, a bounded set A ⊂ X is relatively p-compact if and only if the evaluation map U∗ A : X∗ −→ ∞(A) is p-summing. We obtain p-compactness criteria valid for Banach spaces in Cp.