Question

for any operator T, show T+T* is self-adjoint and T-T* is skew-adjoint

Answer 1

I know if T+T* is self adjoint, I need to show T+T*=T*+T.
How can I prove T+T*=T*+T

Answer 2

Just for clarity, you want to show T+T* = (T+T*)*

You've basically done the proof for the first one. What does the RHS of your equation imply?

Answer 3

(T+T*)*=T*+T

Answer 4

Yes. Although, I would write an extra step just in case the professor doesn't think it's obvious:

(T+T*)*=((T*)*+T*)* =T*+T

The proof for skew-adjoint is similar. If you want to post it, I can check it if you'd like.

Answer 5

so if T+T* is self-adjoint T+T* must be equal to (T+T*)*?

Answer 6

I know if T is a self-adjoint, T=T*

Answer 7

You're stating the converse, sort of. Also, you have no idea if T is self-adjoint or not. Most of the time, it isn't. All you know is T is an operator and you have T + T*.

So, start with that:

Let T be an operator and T+T* be given. Then:

T+T* = T*+T (since addition is commutative)
= ((T*)* + T)* (using definition of transpose: (A+B)* = A*+B*)
= (T+T*)* (since (T*)* = T by definition of transpose)

And this completes the proof.

The proof for skew-adjoint is almost identical.

Answer 8

T-T*=T*-T
=(T*-T)*
=((T*)*-T*)
=T-T*

NO IDEA...

Answer 9

Write out the equation you're trying to get.

Answer 10

my question is if T-T* is skew adjoint. T-T* should be equal to what?

Answer 11

Definition of skew-adjoint:

A = -A*

Here, your A = T-T*
So, what is the equation?

Answer 12

well... I know that if T is skew adjoint T*=-T, so T=-T*
so T-T*=0?

Answer 13

You're showing T-T* is skew-adjoint, not T. You don't know if T is skew-adjoint. It doesn't have to be. You want T-T* as skew-adjoint. Let A = T-T*. Then, plug it into the definition I gave you. What do you have as your equation?

Answer 14

T-T*=-T*+T
=(-T*+T)*
=(-T*)*+T*
=-T+T*

Answer 15

if that's now how you do it. then idk..

Answer 16

not*

Answer 17

It's not. You made a slight mistake in your second line. Here's a hint:

(Remember: You're trying to show T-T* = -(T-T*)*

Notice T-T* = -(-T+T*)

Go from there.

Answer 18

T-T*=-(-T+T*)
=-(T*-T)*
=-((T*)*-T*)
=-(T-T*)

Answer 19

=-(T*-T)*
=-((T*)*-T)*
=-(T-T)*

Answer 20

Close. You made an error distributing the transpose on the 2nd line of your first reply. But, you have the right idea. (Remember: A* + B = (A + B*)*

I need to go to bed. I think you should be able to do it from here without a problem though. Have a good night.

Answer 21

k. Thanks for your help!!!