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Homework Statement
Let V be a complex inner product space, and let T be a linear operator on V.
Define
T_1 = 1/2 (T + T*) and T_2 = (1/2i)(T – T*)
a) Prove that T_1 and T_2 are self-adjoint and that T = T_1 + T_2
b) Suppose also that T = U_1 + U_2, where U_1 and U_2 are self-adjoint. Prove that U_1 = T_1 and U_2 = U_2.
c) Prove that T is normal iff (T_1)(T_2) = (T_2)(T_1)
Homework Equations
Self-adjoint: T = T*
Normal: TT* = T*T
The Attempt at a Solution
(a) was very easy. However, I cannot get started on (b) at all.
For (c)
One direction: assume (T_1)(T_2) = (T_2)(T_1), show T is normal
Substitute:
(1/2 (T + T*))((1/2i)(T – T*)) = ((1/2i)(T – T*))( 1/2 (T + T*))
(1/4i)(T^2 – T* ^2) = (1/4i)(T^2 – T*^2)
I’m not sure what to do now, or if it’s even the right direction.
Thanks for your help!