I've been asked to prove in a homework problem exactly what the title describes. This was the $3^{rd}$ part of a question whose first 2 parts were to prove that $\ker T=\ker TT^*$ and $\ker T=\ker T^n$, but I can't find any way to implement these into this part. I've also tried proving that
$$\operatorname{Tr}[(P_{w^\bot}TP_w)(P_{w^\bot}TP_w)^*]=0$$ where $P_w$ is the projection onto the eigenspace, but I've reached the expression
$$\operatorname{Tr}[(P_{w^\bot}TP_w)(P_{w^\bot}TP_w)^*]=\operatorname{Tr}(P_wTP_wT)-\operatorname{Tr}(TP_wT)$$ and got stuck there. I would appreciate assistance.
Thanks in advance.
Suppose $\lambda$ is an eigenvalue of $T+T^*$ and $V_{\lambda}$ is its eigenspace. We have to show that if $v\in V_{\lambda}$ then $T(v)\in V_{\lambda}$ as well. And indeed:
$(T+T^*)(Tv)=(T^2+T^*T)(v)=(T^2+TT^*)(v)=T(T+T^*)(v)=T(\lambda v)=\lambda T(v)$
The equality $T^*T=TT^*$ is true because $T$ is normal.