A remark on the approximation of p-compact operators by finite-rank operators

The Banach operator ideal Kp of p-compact operators was introduced in [P.D. Sinha, A.K. Karn, Compact operators whose adjoints factor through subspaces of ℓp, Studia Math. 150 (2002) 17–33]. Let p⩾1 and let X be a closed subspace of an Lp(μ)-space. We show that X has the approximation property if and only if for every Banach space Y, the linear space F(Y,X) of finite-rank operators is Kp-norm dense in Kp(Y,X), i.e., X has the Kp-approximation property.