The (E, E)-approximation property for Banach spaces

For a given Banach space E with an 1-unconditional basis, we introduce the (E, E)-approximation property ((E, E)-AP) which is a natural generalization of the classical approximation property (AP) and the (p, p)-AP. It is shown that the AP always implies the (E, E)-AP. If E has an 1-unconditional and shrinking basis, then the (E*, E*)-AP for X* implies the (E, E)-AP for X. If in addition that the unconditional basis for E is subsymmetric, then X* has the (E*, E*)-AP implies that X has the duality (E, E)-AP.