Bounded approximation properties via integral and nuclear operators

Let $X$ be a Banach space and let $\mathcal A$ be a Banach operator ideal. We say that $X$ has the $\lambda$-bounded approximation property for $\mathcal A$ ($\lambda$-BAP for $\mathcal A$) if for every Banach space $Y$ and every operator $T\in \mathcal A(X,Y)$, there exists a net $(S_\alpha)$ of finite rank operators on $X$ such that $S_\alpha\to I_X$ uniformly on compact subsets of $X$ and

$$\limsup_\alpha\Vert TS_\alpha\Vert _{\mathcal A}\leq\lambda\Vert T\Vert _{\mathcal A}.$$

We prove that the (classical) $\lambda$-BAP is precisely the $\lambda$-BAP for the ideal $\mathcal I$ of integral operators, or equivalently, for the ideal ${\mathcal{S{\kern -0.15em}I}}$ of strictly integral operators. We also prove that the weak $\lambda$-BAP is precisely the $\lambda$-BAP for the ideal $\mathcal N$ of nuclear operators.