LAUSR

We investigate the ideal Kup, 1 ≤ p ≤ ∞, of unconditionally p-compact operators. We obtain the isometric identities Kup = Kup ◦ Kup, Kmax up = Lsur p∗ , K min up = ⊗̂/wp∗ and Kup = N up and prove that, if X∗ has the approximation property or Y has the Kup-approximation property, then Kup(X,Y ) is isometrically equal to Kmin up (X,Y ), and the dual space Kup(X,Y )∗ is isometric to (L p )∗(X∗, Y ∗). As a consequence, for every Banach space X, we obtain the isometric identities Kmax up (l1(Γ), X) = Lp∗ (l1(Γ), X), Kmin up (l1(Γ), X) = l∞(Γ)⊗̂wp∗X and Kup(l1(Γ), X) ∗ = Dp∗ (l∞(Γ), X∗).