$T^2$ normal operator means that $T$ is normal operator?

Question

Let $T:V \rightarrow V$ linear transformation, $V=\mathbb{C}$

Is that true that if $T^2$ is normal operator so $T$?

I know that if $T$ is normal so $T^2$ is normal but I have no idea how to think on the opposite direction.

Answer 1

If you take $$ T = \pmatrix{0&1\\0&0}, $$ then $T^2$ is normal but $T$ is not.