1. May 7, 2007

### redyelloworange

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
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.

2. May 7, 2007

### StatusX

For b, you must mean

"Suppose also that T = U_1 + i U_2, where U_1 and U_2 are self-adjoint. Prove that U_1 = T_1 and U_2 = T_2."

This shouldn't be too hard.

3. May 8, 2007

### HallsofIvy

Staff Emeritus
Are you sure of this? It doesn't look to me like T_2 is self adjoint and it is easy to see that T_1+ T_2 is NOT T! Did you mean T= T_1+ i T_2? This is a lot like breaking ex into cosine and sine but, again, T_2 does not satisfy <T_2 u, v>= <u, T_2 v>. It satisfies <T_2u, v>= -<u, T_2 v>.

As StatusX said, this should be U_2= T_2.

4. May 8, 2007