The ideal of weakly p-compact operators and its approximation property for Banach spaces

Abstract

We investigate the ideal Wp of weakly p-compact operators and its approximation property (Wp-AP). We prove that Wp =Wp ◦Wp and Vp = Kup ◦W −1 p and that for 1 < p ≤ ∞, a Banach space X has the Wp-AP if and only if the identity map on X is approximated by finite rank operators on X in the topology of uniform convergence on weakly p-compact sets. Also, we study the Wp-AP for classical sequence spaces and dual spaces.