(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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)

2. Relevant equations

Self-adjoint: T = T*

Normal: TT* = T*T

3. 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!

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# Homework Help: Self-adjoint problem

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