SOME RESULTS OF THE -APPROXIMATION PROPERTY FOR BANACH SPACES

Given a Banach operator ideal $\mathcal A$ , we investigate the approximation property related to the ideal of $\mathcal A$ -compact operators, $\mathcal K_{\mathcal A}$ -AP. We prove that a Banach space X has the $\mathcal K_{\mathcal A}$ -AP if and only if there exists a λ ≥ 1 such that for every Banach space Y and every R ∈ $\mathcal K_{\mathcal A}$ ( Y , X ), $$ \begin{equation} R \in \overline {\{SR : S \in \mathcal F(X, X), \|SR\|_{\mathcal K_{\mathcal A}} \leq \lambda \|R\|_{\mathcal K_{\mathcal A}}\}}^{\tau_{c}}. \end{equation} $$ For a surjective, maximal and right-accessible Banach operator ideal $\mathcal A$ , we prove that a Banach space X has the $\mathcal K_{(\mathcal A^{{\rm adj}})^{{\rm dual}}}$ -AP if the dual space of X has the $\mathcal K_{\mathcal A}$ -AP.