Solving Self-Adjoint Problem in Complex Inner Product Space

[redyelloworange]

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!

 

[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.

 

[HallsofIvy]

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

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 e^x 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>.

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.

As StatusX said, this should be U_2= T_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!

 

[redyelloworange]

Sorry about that,

It should be

T = T_1 + iT_2

T_2 is self adoint: T_2* = (-1/2i)(T*-T) = (1/2i)(T-T*) = T_2