Density of finite rank operators in the Banach space of p-compact operators

A Banach space X is said to have the k p -approximation property ( k p -AP) if for every Banach space Y, the space F ( Y , X ) of finite rank operators is dense in the space K p ( Y , X ) of p-compact operators endowed with its natural ideal norm k p . In this paper we study this notion that has been previously treated by Sinha and Karn (2002) in [15]. As application, the k p -AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi p-nuclear operators for the p-summing norm. This allows to obtain a relation between the k p -AP and Sapharʹs approximation property. As another application, the k p -AP is characterized in terms of a trace condition. Finally, we relate the k p -AP to the ( p , p ) -approximation property introduced in Sinha and Karn (2002) [15] for subspaces of L p ( μ ) -spaces.