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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.

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No. The operator $T$ can vanish on $X$ while any such $S^{**}$ necessarily has its range contained in $X$.

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