Proving a property for a normal transforation $T$ for which $T^{-1}=-T$

Question

Let $V$ be a unitary space.

Given a normal transforation $T$ for which $T^{-1}=-T$.

Let $v \in V$ and $u=Tv$

I need to prove that $Tu=-v$ (which I managed to do easily, so we can consider it as given) and that

$$||T^*u-v||^2+||T^*v+u||^2=0$$

($T^*$ means $T$ traspose and conjugate)

I tried disassambling the inner product using the linear property, but I got to a dead end each time.

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EDIT: Notice that $T$ is normal.

Answer 1

Hint: to show that the sum of two norms is $0$, you need each norm to be zero (because both are nonnegative).