Information about $T$ if $T^2=0$ What can we say about $T$ if $T^2=0$? How can we prove that it has zero as an eigenvalue?

If  $T \ne 0$, then there is a non-zero vector $\mathbf v$ such that $T\mathbf v \ne 0$, and then $T(T\mathbf v) = T^2\mathbf v = 0$, showing that $0$ is an eigenvalue of $T$ with eigenvector $T\mathbf v$.  In fact every eigenvalue of $T$ is $0$, since if $T \mathbf v = \lambda \mathbf v$ for nonzero $\mathbf v$, then $0 = T^2\mathbf v = \lambda^2 \mathbf v$, so since $\mathbf v \ne 0$, $\lambda^2 = 0$, whence $\lambda =0$ as well.
More can be said about $T$:  for example, $I \pm T$ is invertible, since $(I + T)(I -T) = I^2 - T^2 = I$.    The list of "what we can say about T" can be on the long side, so having shown $0$ is an eigenvalue, per specific request, I'll leave further facts for those who seek them to discover.
Hope this helps.  Cheers,
and as always,
Fiat Lux!!!
