Can Any Linear Operator Be Expressed Using Hermitian Components?

[andre220]

Homework Statement

Show that any linear operator $\hat{L}$ can be written as $\hat{L} = \hat{A} + i\hat{B}$, where $\hat{A}$ and $\hat{B}$ are Hermitian operators.

Homework Equations

The properties of hermitian operators.

The Attempt at a Solution

I am not sure where to start with this one. For example, we know that if an operator, A is hermitian, then $\langle g\mid A f \rangle = \langle f\mid A g\rangle^*$. But I do not see how to break up L into any combination of other operators. Any help would be appreciated, perhaps a nudge in the right direction.

 

[strangerep]

Show that any linear operator $\hat{L}$ can be written as $\hat{L} = \hat{A} + i\hat{B}$, where $\hat{A}$ and $\hat{B}$ are Hermitian operators.

What would $\hat L^\dagger$ look like?

 

[Matterwave]

If $\hat{A},\hat{B}$ are Hermitian then $i\hat{B}$ is anti-Hermitian. So the problem is really just asking you to prove that any operator is a sum of a Hermitian part and an anti-Hermitian part. This is very similar that any real linear operator is a sum of a symmetric part and an anti-symmetric part. Do you know how that property is proved?