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

The Banach operator ideal K p of p-compact operators was introduced in [P.D. Sinha, A.K. Karn, Compact operators whose adjoints factor through subspaces of ℓ p , Studia Math. 150 (2002) 17–33]. Let p ⩾ 1 and let X be a closed subspace of an L p ( 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 K p -norm dense in K p ( Y , X ) , i.e., X has the K p -approximation property.