$\ell_p$-strictly singular operators and non-$\ell_p$-factoring operators

Question

Let $X,Y,E$ be Banach spaces. We say that a continuous linear operator $T\in\mathcal{L}(X,Y)$ is $E$-strictly singular, and write $T\in\mathcal{SS}_E(X,Y)$, whenever it fails to fix a copy of $E$. More precisely, $T\in\mathcal{SS}_E(X,Y)$ whenever for each closed subspace $U\subseteq X$ satisfying $U\approx E$, the operator $T$ is not bounded below on $U$, i.e. for $\epsilon>0$ there is $u\in U$ such that $\|Tu\|<\epsilon\|u\|$.

We say that $T$ is non-$E$-factoring, and write $T\in\mathcal{M}_{\ell_p}(X,Y)$, whenever $Id_E$ fails to factor through $T$, i.e. whenever there are no $A\in\mathcal{L}(E,X)$ and $B\in\mathcal{L}(Y,E)$ such that $BTA=Id_E$, where $I_E\in\mathcal{L}(E)$ is the identity operator on $E$.

It is clear that we always have $\mathcal{SS}_E(X,Y)\subseteq\mathcal{M}_E(X,Y)$. I believe (but have not yet checked to be sure) that these classes coincide whenever $E=\ell_1$ or $E=c_0$. If $1<p<\infty$, we also have $\mathcal{SS}_{\ell_p}(L_p)=\mathcal{M}_{\ell_p}(L_p)$ and $\mathcal{SS}_{\ell_2}(L_p)=\mathcal{M}_{\ell_2}(L_p)$.

Question 1. Let $1<p<\infty$. Do there exist Banach spaces $X$ and $Y$ such that $\mathcal{SS}_{\ell_p}(X,Y)\neq\mathcal{M}_{\ell_p}(X,Y)$?

It is known that $\mathcal{SS}_{c_0}$ and $\mathcal{SS}_{\ell_p}$ are norm-closed operator ideals for $1\leq p<\infty$. I believe (but have not checked) that the same is true for $\mathcal{M}_{c_0}$ and $\mathcal{M}_{\ell_1}$. I believe (but have not checked) that $\mathcal{M}_{\ell_p}$ is also an operator ideal in the sense of Pietsch for $1<p<\infty$, but I do not know whether it is norm-closed.

Question 2. Let $1<p<\infty$. Do there exist Banach spaces $X$ and $Y$ such that $\mathcal{M}_{\ell_p}(X,Y)$ fails to be norm-closed in $\mathcal{L}(X,Y)$ (in the operator norm)?

An example satisfying Q2 would also satisfy Q1.

One last thing: Are there references for any of the "known" facts I mentioned above? For example, where is it proved that $\mathcal{SS}_{\ell_p}$ is a norm-closed operator ideal?

Thanks guys!

Answer 1

I think that it is still open whether $\mathscr{M}_{\ell_p}$ is closed under addition, hence it is not clear whether is coincides with $\mathscr{S\!\!S}_{\ell_p}$.

That $\mathscr{S\!\!S}_{c_0(\Gamma)}$ is a closed operator ideal (for any set $\Gamma$) is not too hard to show. See Proposition 3.13 in

W. B. Johnson, T. Kania and G. Schechtman, Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support, Proc. Amer. Math. Soc. 144 (2016), 4471-4485.

That $\mathscr{S\!\!S}_E$ is a closed operator ideal for a minimal space $E$ follows from a variation of Kato's lemma; see Proposition 2.5 in

T. Kania and N. J. Laustsen, Uniqueness of the maximal ideal of operators on the $\ell_p$-sum of $\ell_\infty^n$ ($n\in \mathbb N$) for $1<p<\infty$, Math. Proc. Camb. Phil. Soc. 160 (2016) 413-421.