# A question on $p$-approximation property

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 $\prod_{p}$ ?

• Typographical note: Use $\langle x^*,x\rangle$ (with \langle and \rangle) rather than $<x^*,x>$. For me at least, the second formatting is difficult to read. (I made this edit.) Commented Aug 27, 2014 at 9:16

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.

• Thank you for your answer. But I can not read your master's thesis because it is not written in English. Commented Dec 12, 2015 at 15:11
• Sorry about that, I regret not having written in English. I'll try to translate the mentioned results and post here. Commented Dec 18, 2015 at 12:55
• Well, I translated (very fast, sorry for possible errors in my english) the statements, if you need the translation of some of the demonstrations, let me know. Commented Dec 18, 2015 at 13:14