find all eigenvalues for this specific transformation

Question

Let $V$ be an inner-product space, and $T:V \to V$ a linear transformation for which:
$T^2 = \frac{T+T^*}{2}$

What are all $T$'s eigenvalues? ($T^*$ notates the conjugate transpose of $T$).

Thanks for any help!

Answer 1

Note that $T^* = 2T^2 - T$. Conclude that $T$ is normal, i.e. $TT^* = T^*T$.

By the spectral theorem for normal matrices, conclude that $T$ is unitarily diagonlizable. It follows that because $T^2 = \frac{T + T^*}{2}$, every eigenvalue $\lambda$ of $T$ must satisfy $$ \lambda^2 = \frac{\lambda + \bar \lambda }{2} \implies \lambda^2 = \operatorname{Re}[\lambda]. $$ What are the solutions to this equation?