showing an operator is normal If I have that H is an inner product space with inner product $( . , . )$ over the complex numbers, and $T∈L(H,H)$.  Let $R=T+T^*$, $S=T-T^*$ . 
If I suppose T is normal, how do I show that :
1) $T^*$ is normal and
2) $R∘S=T∘T-T^*∘T^*$
I'm having trouble even getting started on this problem.  I appreciate any and all help.  Thanks!

Definitions:


*

*T normal <=> TT* = T*T

*T* normal <=> T*T** = T**T*
Furthermore we have the following properties of the adjoint:


*

*T** = T

*(R + S)* = R* + S*
can you take it from there?
