We say that a subset $K$ of a Banach space $X$ is relatively $p$-compact ($1\leq p<\infty$) if there exists a $p$-summable sequence $(x\_n)\_{n=1}^{\infty}$ in $X$ such that 
$$ K\subseteq \left\{\sum\_{n=1}^{\infty}\alpha\_{n}x\_{n}:(\alpha\_{n})\_{n}\in B\_{l\_{q}}\right\}\quad\quad(\frac{1}{p}+\frac{1}{q}=1).$$
 A Bananch space $X$ is said to have the $p$-approximation property if $I\_{X}$ can be approximated by finite rank operators uniformly on the relatively $p$-compact subsets of $X$. An operator $T:X \rightarrow Y$ is said to be quasi $p$-nuclear if there exists a $p$-summable sequence $(x^{\*}\_{n})\_{n=1}^{\infty}$ in $X^{\*}$ such that
$$\|Tx\|\leq \left(\sum\_{n=1}^{\infty}|\langle x^{\*}\_{n},x\rangle|^{p}\right)^{1/p} \quad\quad\hbox{ for all $x\in X$.} $$


We denote the Banach ideal of quasi $p$-nuclear operators by $\mathcal{Q}\mathcal{N}\_{p}$ and the Banach ideal of $p$-summing operators by $\prod\_{p}$. We say that a Banach space $X$ has the right approximation property with respect to $\mathcal{Q}\mathcal{N}\_{p}$ (or $\prod\_{p}$)if for every Banach space $Y$ and every operator $T\in \mathcal{Q}\mathcal{N}\_{p}(X,Y)$(or $T\in \prod\_{p}(X,Y)$), we have $T\in \overline{\{TS:S\in \mathcal{F}(X,X)\}}^{\tau}$, where $\tau$ is the topology of uniform convergence on compact subsets of $X$. My questions are: 


1. If $X$ has the right approximation property with respect to $\mathcal{Q}\mathcal{N}\_{p}$, does $X$ has the $p$-approximation property ?
2. If $X^{\*}$ has the $p$-approximation property, does $X^{\*}$ have the right approximation property with respect to $\prod\_{p}$ ?



A question on $p$-approximation property
The answer for both questions are positive. The first is an easy consequence of a version of Grothendieck's Theorem that characterizes the p-approximation property in terms of the density of finite-rank operators w.r.t uniform convergence on p-compact sets (1). Since p-compact sets are q-compact for p

It follows the number of the results in my thesis I have mentioned above:
(1) Proposition 3.1.8
(2) Proposition 2.2.4
(3) Proposition 3.3.3


Link for my master's thesis: <http://www.teses.usp.br/teses/disponiveis/45/45131/tde-12112013-232741/en.php>


Proposition 3.1.8


Let $X$ be a Banach sapce and $1\leq p \leq \infty$, then are equivalent:


(i) $X$ has the $p$-aproximation property.


(ii) For every Banach space $Y$, $\mathcal{F}(Y,X)$ is $\tau\_p$-dense in $\mathcal{B}(Y,X)$.


Proposition 2.2.4


Let $1 \leq p< r \leq \infty$ and suppose $K$ is a relative $p$-compact set, then $K$ is relative $r$-compact.


Proposition 3.3.3


Let $X$ be a Banach space and $1 \leq p \leq \infty$. If $X^\*$ has the $p$-approximation property then $X$ has the p-approximation property too.




