Let be a Banach operator ideal. Based on the notion of -compactness in a Banach space due to Carl and Stephani, we deal with the notion of measure of non–compactness of an operator. We consider a map ${\chi }_{}$ (respectively, $n_{}$) acting on the operators of the surjective (respectively, injective) hull of such that ${\chi }_{}\left(T\right)=0$ (respectively, $n_{}\left(T\right)=0$) if and only if the operator T is -compact (respectively, injectively -compact). Under certain conditions on the ideal , we prove an equivalence inequality involving ${\chi }_{}(T*)$ and $n_{{}^d}\left(T\right)$....