# A question on the metric approximation property

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.

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