If $T^9=T^8$, prove that $T^2=T$. Suppose $V$ is a complex inner product space and $T $ is a normal operator on $V$ such that $T^9=T^8$. Prove that $T^2=T$. Give an example of a non-normal $T$ such that $T^9=T^8$ but $T^2 \ne T$.

As example for non-normal one just consider any matrix with at least one nilpotent Jordan-block of size 3 or larger. What will happen if we raise those block to 1 or 2 or 3 respectively?
Example of such block:
$$\begin{bmatrix}0&1&0\\0&0&1\\0&0&0\end{bmatrix}$$
