The injective and surjective hulls of the ideal of (p,q)-compact operators and their approximation properties

Abstract

Let 1 ≤ p , q ≤ ∞ with 1 / p + 1 / q ≥ 1 . We establish the isometric representations of the injective hull K i n j p , q and surjective hull K p , q s u r of the ideal of ( p , q ) -compact operators. We obtain some factorizations of K p , q i n j and K p , q s u r , and prove that if a Banach space X has the K p , p ⁎ i n j -approximation property ( K p , p ⁎ i n j -AP) (respectively, K p , p ⁎ s u r -AP), then X has the K p , q i n j -AP (respectively, K p , q s u r -AP). It also follows that if a Banach space X has the AP, then X has the K p , q i n j -AP and the K s u r p , q -AP, and that every Banach space has the K i n j 2 , q -AP, the K p , 2 i n j -AP, K s u r 2 , q -AP and the K p , 2 s u r -AP. Finally, we find the dual space of ( K p , q i n j ( X , Y ) , ‖ ⋅ ‖ K p , q i n j ) (respectively, ( K p , q s u r ( X , Y ) , ‖ ⋅ ‖ K p , q s u r ) ) when X ⁎ has the AP or Y has the K p , q i n j -AP (respectively, K p , q s u r -AP).