HK hermitian if H and K commute

Question

$\text{Let H and K be hermitian operators in unitary space.}$ $\text{HK is hermitian only if HK commutes (HK = KH).}$

How would one prove this statement without the use of eigenvalues?

Answer 1

Suppose $H$, $K$ and $HK$ are hermitian, then $HK = (HK)^\dagger = K^\dagger H^\dagger = KH$.