Let $X,Y$ be Banach spaces. Suppose that $X^{\*\*\*}$ has the metric approximation property. Let $T:X^{\*\*}\rightarrow Y$ be a finite-rank operator and let $\epsilon>0$. Is there a finite-rank operator $S:X\rightarrow X$ such that $\|S\|\leq 1+\epsilon$ and $\|T-TS^{\*\*}\|<\epsilon$?


Furthermore, I want to know whether there are other conditions on $X$ to ensure that this question holds true. 



A question on the metric approximation property
No. The operator $T$ can vanish on $X$ while any such $S^{\*\*}$ necessarily has its range contained in $X$. 


