Yes/no question about non-commuting Hermitian operators

nomadreid
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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
 
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If A and B are Hermitian, then AB-BA is anti-Hermitian, (##M^\dagger = - M##). ##i(AB-BA)## is therefore Hermitian.
 
fzero: Thank you very much.
 

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