Can non-commuting Hermitian operators have commutator equal to a Hermitian operator?

[nomadreid]

Is the following a theorem? yes or no
If A and B are non-commuting Hermitian operators (or matrices), there does not exist Hermitian operators C and D such that AB-BA = CD.
(Or, as special case, ...there does not exist a Hermitian operator C s.t. C= AB-BA)
Thanks

 

[fzero]

If A and B are Hermitian, then AB-BA is anti-Hermitian, ($M^\dagger = - M$). $i(AB-BA)$ is therefore Hermitian.

 

[nomadreid]

fzero: Thank you very much.