Davis, Figiel, Johnson and Pełczyński factorization through spaces with a bases 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.
 A: 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.
