# Is this operator Hermitian?

• sgoodrow

#### sgoodrow

1. Let G be an operator on H (Hilbert Space). Show that:
(a) H = 1/2 (G + G$$^{\dagger}$$) is Hermitian.
(b) K = -1/2 (G - G$$^{\dagger}$$) is Hermitian.
(c) G = H + iK
.

## Homework Equations

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3. The Attempt at a Solution :

(a) Since the adjoint of the sum of two operators does not change their position (addition of operators is commutative), it is very straight forward.

(b) This is where I run into trouble, and I think it is because the problem is wrong. The operator given appears to be anti-hermitian (skew-hermitian), however I would like confirmation. This result makes (c) very difficult, as it uses an incorrect answer. My suspicion is that the intended question includes an i (as in, H = -i/2 (...)):

K$$^{\dagger}$$ = -1/2 (G$$^{\dagger}$$ - G) = 1/2 (G - G$$^{\dagger}$$) = -K.

(c) Well if I am correct about (b), (c) is wrong. I was just hoping someone could confirm I am right, or show me how K is Hermitian. I can take it from there.

If the problem should be as stated this will give a condition for G that allows to solve all equations.
If G is arbitrary and the equations should be satisfied for general G there has to be a change in definition.
K= -i/2(...)