Prove that $v = 0 \iff v + iT(v) = 0$ for a self adjoint operator

Question

Let $T:V\rightarrow V$ be a self adjoint operator over $C$. Prove that $v = 0 \iff v + iT(v) = 0$.

I've tried to use a norm and equate it to $0$, and also to use $\langle v + iT(v),u\rangle = 0$ for every $u\in V$, but i always get stuck. Maybe there's something very basic i'm missing here.

Answer 1

Hint The "$\implies$" direction is easy. For the other direction, go by contrapositive. Suppose that $v + iT(v) = 0$. That is, we have $v = -iT(v) \implies T(v) = iv$.

Use the fact that $T$ is self-adjoint to show that $v = 0$.