Applications of the result : $T$ locally algebraic $\Rightarrow$ $T$ algebraic

Question

I came across the following result some time ago and I was wondering if it had any "nice" applications:
If $E$ is a Banach space and $T:E\rightarrow E$ is a continuous linear operator which is locally algebraic (i.e. $\forall x\in E$ there is a nonzero polynomial $p(t)$ such that $p(T)(x)=0)$ then $T$ is algebraic (i.e. there is a nonzero polynomial $p(t)$ such that $p(T)=0)$
I would be grateful if you could give me some applications.