  We use the notion of $\A$-compact sets, which are determined by a Banach
operator ideal $\A$, to show that most classic results of certain approximation
properties and several Banach operator ideals can be systematically studied
under this framework. We say that a Banach space enjoys the $\A$-approximation
property if the identity map is uniformly approximable on $\A$-compact sets by
finite rank operators. The Grothendieck's classic approximation property is the
$\K$-approximation property for $\K$ the ideal of compact operators and the
$p$-approximation property is obtained as the $\mathcal N^p$-approximation
property for $\mathcal N^p$ the ideal of right $p$-nuclear operators. We
introduce a way to measure the size of $\A$-compact sets and use it to give a
norm on $\K_\A$, the ideal of $\A$-compact operators. Most of our results
concerning the operator Banach ideal $\K_\A$ are obtained for right-accessible
ideals $\A$. For instance, we prove that $\K_\A$ is a dual ideal, it is regular
and we characterize its maximal hull. A strong concept of approximation
property, which makes use of the norm defined on $\K_\A$, is also addressed.
Finally, we obtain a generalization of Schwartz theorem with a revisited
$\epsilon$-product.

Functional Analysis