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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 $X\subseteq Y$ 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 $\ell_{1}(X)$.

  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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It is "Davis", not "Davies". –  Bill Johnson Aug 10 '10 at 15:52
    
Thanks for the correction. –  G. Rodrigues Aug 10 '10 at 16:09
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2 Answers

I think the most user friendly introduction to the topic is the book of Diestel, Jarchow, and Tonge, so I suggest you locate that book. There you will find the description of the dual ideal to the operators that factor through some $L_1$ space. (Actually it is better to look at operators $T:X\to Y$ such that $i_YT$ factors through some $L_1$ space, where $i_Y$ is the canonical embedding of $Y$ into $Y^{**}$. This operator ideal is called $\Gamma_1$.) You can also find the basic things in N. Tomczak-Jaegermann's book ``Banach-Mazur distances and finite-dimensional operator ideals", published by Longman.

A subproblem to characterizing the operators which factor through $\ell_1$ is to characterize the operators into $L_1$ that factor through $\ell_1$. I think this has not been done. The operators FROM $L_1$ that factor through $\ell_1$ are the differentiable operators (D.R. Lewis and C. Stegall, JFA 12 (1973)).

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Since you mention the Lewis-Stegall paper, I presume that what you call "differentiable operators" are the representable operators, that is, those $T:L_{1}(\Omega)\to B$ for which there is a measurable a.e.-bounded $g:\Omega\to B$ such that $Tf= \int gf$, correct? –  G. Rodrigues Aug 10 '10 at 16:11
    
Well, yes, but differentiable usually refers to an equivalent form. One of the other nice equivalences is that every set of positive measure contains a smaller set of positive measure s.t. the restriction of the operator to the functions supported on the smaller set is a compact operator. –  Bill Johnson Aug 10 '10 at 18:13
    
Just thought I'd mention that if the OP doesn't have access to a library carrying a hardcopy, or know someone who owns one, then (s)he's unlikely to find N. Tomczak-Jaegermann's book, as it's near impossible to get hold of a copy any other way. But obviously that won't matter if (s)he can get hold of Diestel, Jarchow and Tonge's book, much of which can be viewed on Google Books: books.google.com.au/… –  Philip Brooker Aug 10 '10 at 21:09
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In the paper R. Cilia and J. M. Gutierrez, Left $\ell_1$-factorable polynomials, Glasgow Math. J. 51 (2009), 631-649 you may find some results on operators factoring through $\ell_1$

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