The ideal of unconditionally p-compact operators

We investigate the ideal K ₔ₏, 1 p, of unconditionally p-compact operators. We obtain the isometric identities K ₔ₏= K ₔ₏ K ₔ₏, K^ ₔ₏= L^ sur₏^*, K^ ₔ₏= /ₖ_₏^* and K ₔ₏= N ₔ₏^ Qdual and prove that, if X^* has the approximation property or Y has the K ₔ₏-approximation property, then K ₔ₏ (X, Y) is isometrically equal to K^ ₔ₏ (X, Y), and the dual space K ₔ₏ (X, Y) ^* is isometric to (L₏^ inj) ^* (X^*, Y^*). As a consequence, for every Banach space X, we obtain the isometric identities K ₔ₏^ (₁ (), X) = L₏^* (₁ (), X), K ₔ₏^ (₁ (), X) = () ₖ_₏^* X and K ₔ₏ (₁ (), X) ^* = D₏^* ( (), X^*).