Proving the adjoint nature of operators using Hermiticity

Question

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?

Answer 1

Hint:

For two Hemitian operators $A= A^\dagger$ and $B=B^\dagger$ and $a$ a real number you have:

$$ (A+aiB)^\dagger=A^\dagger+(aiB)^\dagger=A^\dagger-aiB^\dagger=A-aiB $$

You can see here.