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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 $n_{{}^d}\left(T\right)$....

Given an operator ideal , we say that a Banach space X has the approximation property with respect to if T belongs to ${\overline{S\circ T:S\in ℱ\left(X\right)}}^{{\tau }_c}$ for every Banach space Y and every T ∈ (Y,X), ${\tau }_c$ being the topology of uniform convergence on compact sets. We present several characterizations of this type of approximation property. It is shown that some of the existing approximation properties in the literature may be included in this setting.