Let $A$ be a Banach algebra and $A^{ **}$ be the second dual of it. We show that by some new conditions, $A$ is weakly amenable whenever $A^{ **}$ is weakly amenable. We will study this problem under generalization, that is, if $(n+2)-th$ dual of $A$, $A^{ (n+2)}$, is $T-S-$weakly amenable, then $A^{ (n)}$ is $T-S-$weakly amenable where $T$ and $S$ are continuous linear mappings from $A^{ (n)}$ into $A^{ (n)}$.
Categories: math.FA