Show that $B(X)$ is a semisimple Banach algebra, where $X$ is a Banach space. That is, to show that rad $B(X)=\{0\}$, or equivalently, to show $\sigma(AT)={0} \, \forall T\in B(X)\Rightarrow A=0$.
I have been trying to find a simple proof of the following without using the more general result that for a Banach space $X$, $\sigma(AT)=\sigma(BT) \, \forall T\in B(X)\Rightarrow A=B$.