If I take the anti-commutator of two positive operators $A,B$ on a Hilbert space, $AB+BA$ is again guaranteed to be Hermitian, but is it also necessarily positive?
Positivity of the anti-commutator of two positive operators
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linear-algebra
functional-analysis
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1Actually, I just found a counter-example using 2-dimensional complex matrices, so please ignore this question. So, the answer is NO, in general the anti-commutator need not be positive. – 2012-01-28
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0$A=\begin{bmatrix}1&0\\0&0\end{bmatrix}$ and $B=\begin{bmatrix}1&1\\1&1\end{bmatrix}$ gives a counterexample. You could post an answer to your own question if you want to. – 2012-01-28