Self Adjoint and Anti-Self Adjoint questoin

[Alupsaiu]

I'm having trouble showing that any normal linear transformation T is the sum of a self-adjoint transformation T1 and anti-self adjoint linear transformation T2, (so T=T1+T2) so that T1 and T2 commute. Anti-self adjoint being <Ta,b>=-<a,Tb>.
Specifically I'm not sure how to use the information of T being normal.
Any help is appreciated, thank you.

 

[micromass]

Hi Alupsaiu!

What about this standard trick:

$$T=\frac{T+T^*}{2}+\frac{T-T^*}{2}$$

 

[Alupsaiu]

...Moments like these I feel so small...haha thanks a bunch