Proving the adjoint nature of operators using Hermiticity

[spaghetti3451]

How can the fact that $\hat x$ and $\hat p$ are Hermitian be used to prove that $\hat x - \frac{i}{m \omega} \hat p$ and $\hat x + \frac{i}{m \omega} \hat p$ are adjoints of each other?

 

[RUber]

Adjoints have the property that $\langle A, u \rangle =\langle u, A^* \rangle$. In the simplest case, assume that x and p are purely real. Then you should be able to show that the conjugate matrices satisfy the required properties.

 

[Fredrik]

Prove the rules $(A+B)^\dagger =A^\dagger+B^\dagger$ and $(cA)^\dagger=c^*A^\dagger$, and then use them.

Hint: $\langle x,Ay\rangle =\langle A^\dagger x,y\rangle$ for all $x,y$ such that $y$ is in the domain of A, and $x$ is in the domain of $A^\dagger$.