Recently, I bumped into the class of operators that factor through $\ell_{1}(X)$ for some set $X$. For now, $X$ is a set with arbitrary cardinality but if it leads to a more concrete answer to my questions below, feel free to restrict $X$ to be countable. The restriction makes little to no difference to what it follows. I should also warn that I am fairly ignorant of operator ideals and Banach space theory, so please be gentle.
First some definitions. Define,
$$ \|T\|_{\ell_{1}}= \inf\{\|R\|\|S\|\} $$
where the infimum is taken over all the factorizations of $T$ as $\xrightarrow{S}\ell_{1}(X)\xrightarrow{R}$. Obviously, $\|T\|\leq \|T\|_{\ell_{1}}$. Define $\mathcal{L}(A, B)$ to be the linear space of operators that factor through some $\ell_{1}(X)$ with the above norm. The little bit of thought that I have dedicated to this has produced the following up to now (and please correct me, if I have fumbled somewhere):
We have $\|RTS\|_{\ell_{1}}\leq \|R\|\|T\|_{ell_{1}}\|S\|$. Sketch: obvious.
The normed space $\mathcal{L}(A, B)$ is complete. Sketch: If $(T_{n})$ is $\|,\|_{\ell_{1}}$-Cauchy then it has a uniform limit. To prove that this limit factors through some $\ell_{1}(X)$ note two things. First, if you have a factorization through $\ell_{1}(X)$ as $RS$ and $X\subseteq Y$ then, since $\ell_{1}(X)$ is a norm-1 complemented subspace of $\ell_{1}(Y)$, you can make the factorization to pass through the larger $\ell_{1}(Y)$ without altering $\|R\|\|S\|$. Second, one has the isometric isomorphism, $$ \sum_{n}\ell_{1}(X_{n})\cong \ell_{1}(\coprod_{n} X_{n}) $$ which allows to take a sequence of factorizations and push them all to a common space $\ell_{1}(X)$. Thus the uniform limit factors through some $\ell_{1}(X)$.
Finite-rank operators factor through $\ell_{1}(X)$. Sketch: all finite-dimensional spaces are linearly homeomorphic to $\ell_{1}(n)$. These first three conditions taken together mean that $(A, B)\mapsto \mathcal{L}(A, B)$ is an operator ideal (or Banach ideal, I am uncertain of the official terminology).
Each $T$ is completely continuous. Sketch: a sequence in $\ell_{1}(X)$ lives inside a copy of $\ell_{1}$. The Schur property of $\ell_{1}$ gives the result.
Now for my first batch of questions: can this class of operators be characterized? Any more salient properties of these operators? And what about the norm $\|,\|_{\ell_{1}}$, is there some other more enlightening description of it? How far is it from the operator norm?
The second batch of questions is related to what are the properties required of a full subcategory $C$ of the category of Banach spaces so that one obtains an operator ideal by factorizing operators through it. An obvious example is the ideal of weakly compact operators that by Davis-Figiel-Johnson-Pelczynski is the class of operators that factor through reflexive spaces. My guess is that something like $\omega_{1}$-filteredness of $C$ with $\omega_{1}$ the first uncountable ordinal, is enough for the argument to go through, but I am sure someone smarter and more knowledgeable has already thought about this.
If you have appropriate references, that would be great; extra kudos if available online. Next September I will have access to a library and plan to get my hands on the Defant, Floret monograph Tensor norms and operators ideals -- not a very cheerful prospect actually, as the book looks rather daunting. The book Absolutely summing operators by Diestel, Jarchow and Tonge should also be useful, but alas, last time I checked it was not available.
Regards, TIA, G. Rodrigues
