fa.functional analysis - Factorization through $ell_{1}$ and operator ideals

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):

1. We have $|RTS|_{ell_{1}}leq |R||T|_{ell_{1}}|S|$. Sketch: obvious.




2. 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 $Xsubseteq Y$ 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 $ell_{1}(X)$.




3. 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).




4. 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

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