Let $F$ and $H$ be normed spaces and let $E$ be a locally convex space. Let $T:F\to H$ and $S:H\to E$ be linear operators, such that $\|T\|= 1$, $S$ is an injective semi-embedding (i.e. $S\overline{B}_{H}$ is closed in $E$), and $ST$ is weakly compact.
Does it follow that $T$ is weakly compact?
The intuition is that if we consider $F^{**}\to ^{T^{**}} H^{**}\to^{S^{**}} E^{**}$, then $S^{**}T^{**}\overline{B}_{F^{**}}$ is equal to the closure of $ST\overline{B}_{F}$ in $E$, due to weak compactness of $ST$. Since $\|T\|=1$, it follows that $ST\overline{B}_{F}\subset S\overline{B}_{H}$, which is closed in $E$, and so $S^{**}T^{**}\overline{B}_{F^{**}}\subset S\overline{B}_{H}$. Hence, perhaps it is possible to show that $T^{**}\overline{B}_{F^{**}}\subset \overline{B}_{H}$, which is equivalent to weak compactness of $T$.
