The ideal of unconditionally p-compact operators

We investigate the ideal Kᵤₚ, 1 ≤ p ≤ ∞, of unconditionally p-compact operators. We obtain the isometric identities Kᵤₚ= Kᵤₚ∘ Kᵤₚ, Kᵐᵃˣᵤₚ= Lˢᵘʳp^*, Kᵐⁱⁿᵤₚ= ⊗ _/wp^* and Kᵤₚ= NᵤₚQdual and prove that, if X^* has the approximation property or Y has the Kᵤₚ-approximation property, then Kᵤₚ(X, Y) is isometrically equal to Kᵐⁱⁿᵤₚ(X, Y), and the dual space Kᵤₚ(X, Y)^* is isometric to ( Lₚⁱⁿʲ)^*(X^*, Y^*). As a consequence, for every Banach space X, we obtain the isometric identities Kᵤₚᵐᵃˣ( ₁(Γ ), X) = Lp^*( ₁(Γ ), X), Kᵤₚᵐⁱⁿ( ₁(Γ ), X) = ∞(Γ ) ⊗ _wp^* X and Kᵤₚ( ₁(Γ ), X)^* = Dp^* ( ∞(Γ ), X^*).