Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
Actually, 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. – Nikolas Jan 28 '12 at 0:35
$A=\begin{bmatrix}1&0\\0&0\end{bmatrix}$ 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. – Jonas Meyer Jan 28 '12 at 0:37
Thanks Jonas, I'll do that next time. – Nikolas Feb 11 '12 at 21:21

Since the "next time" never came for the OP, I post the counterexample given by Jonas Meyer.

Let $A=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ and $B=\begin{pmatrix}1&1\\1&1\end{pmatrix}$, then $AB+BA=\begin{pmatrix}2&1\\1&0\end{pmatrix}$ has negative determinant.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.