Operators ideals and approximation properties

We use the notion of $\text{A}$-compact sets, which are determined by a Banach operator ideal $\text{A}$, 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 $\text{A}$-approximation property if the identity map is uniformly approximable on $\text{A}$-compact sets by finite rank operators. The Grothendieck’s classic approximation property is the $\text{K}$-approximation property for $\text{K}$ 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 $\text{A}$-compact sets and use it to give a norm on ${\text{K}}_{\text{A}}$, the ideal of $\text{A}$-compact operators. Most of our results concerning the operator Banach ideal ${\text{K}}_{\text{A}}$ are obtained for right-accessible ideals $\text{A}$. For instance, we prove that ${\text{K}}_{\text{A}}$ 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 ${\text{K}}_{\text{A}}$, is also addressed. Finally, we obtain a generalization of Schwartz theorem with a revisited $ϵ$-product.