If I have three operators such that $[A,B] = C$, and I know that $A$ and $C$ are Hermitian, does it follow that $B$ is anti-hermitian? If $A$ and $B$ were Hermitian, $C$ would be anti-Hermitian, so $B$ clearly isn't Hermitian (unless its trivial); but I'm having trouble coming up with a counter example or a proof. (The specific problem I'm interested in has $C$ equal to a nonzero constant, if that helps.)
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