Davis, Figiel, Johnson and Pełczyński's Factorization Theorem states that each weakly compact operator $T:X \to Y$ between Banach spaces $X$ and $Y$ factors through a reflexive Banach space $Z$. In addition, they prove that $Z$ can be constructed to have a Schauder basis if $Y$ has a shrinking basis or an unconditional basis. It was later shown that if $Y=C[0,1]$ then $Z$ can be constructed with a basis.


I am interested generally in positive and negative results regarding when $Z$ can be constructed to have a basis. For example:


Q1: If $Y=L\_1[0,1]$, can $Z$ be constructed to have a basis for any weakly compact $T:X \to Y$?


Q2: Is there an example of a space $Y$ with basis and weakly compact $T:X \to Y$ so that every reflexive $Z$ which $T$ factors through does not have a basis?


I'm also interested to know if anyone is aware of other results similar to the those stated above.



Davis, Figiel, Johnson and Pełczyński factorization through spaces with a bases
I think Q1 has a positive answer. To see this, review how the results are proved in DFJP. You have a weakly compact subset $W$ of $Y$ (the closure of the image of the unit ball of $X$ under a weakly compact operator) and you apply the factorization technique (which we now know is just a real interpolation method applied to the pair $(W, B\_Y)$) to $W$ to obtain a larger weakly compact symmetric subset $C$ of $Y$. Consider the normed space that has $C$ as its unit ball. It is proved that $C$ is weakly compact in in this normed space and hence this normed space is reflexive.


When $Y$ has a (always Schauder) basis, instead of applying the interpolation technique to the original weakly compact set $W$, we enlarged $W$ to a bigger weakly compact set $W\_1$ that is invariant for the partial sum projections associated with the basis for $Y$. Then the interpolation technique, applied to $W\_1$, produces a still larger weakly compact set $C$ that is invariant for the partial sum projections associated with the basis for $Y$. From this it is easy to see that the basis for $Y$ is also a basis for the space that has $C$ as its unit ball. 


For $W\_1$ we used the closed convex hull of the union over $n$ of $P\_n W$, where $(P\_n)$ are the partial sum projections for the basis for $Y$. When the basis is shrinking, we proved that $W\_1$ is weakly compact. Suppose that $Y=L\_1$ and you use the Haar basis for $Y$. Even though the Haar basis is far from being shrinking, the same construction produces a weakly compact set $W\_1$. To see this, recall that a weakly closed set $D$ in $L\_1$ is weakly compact iff for all $\epsilon > 0$ there is $M=M\_\epsilon$ s.t. $D\subset M B\_{L\_2} + \epsilon B\_{L\_1}$ (``uniform integrability") and use the fact that both $B\_{L\_2}$ and $B\_{L\_1}$ are invariant under the Haar basis projections.


