Duality of measures of non--compactness

Let A be a Banach operator ideal. Based on the notion of A-compactness in a Banach space due to Carl and Stephani, we deal with the notion of measure of non-A-compactness of an operator. We consider a map chi(A) (respectively, n(A)) acting on the operators of the surjective (respectively, injective) hull of A such that chi(A) (T) = 0 (respectively, n(A)(T) = 0) if and only if the operator T is A-compact (respectively, injectively A-compact). Under certain conditions on the ideal A, we prove an equivalence inequality involving chi(A)(T*) and n(A)(d) (T). This inequality provides an extension of a previous result stating that an operator is quasi p-nuclear if and only if its adjoint is p-compact in the sense of Sinha and Karn.