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I just faced a obvious-looking inequality, but I didn't manage to prove it. Let $H$ be a finite-dimensional Hilbert space, $M, \rho$ positive operators on $H$, $P$ an orthogonal projector on $H$. Is it true, that $\text{tr}(M \rho) \geq \text{tr}(M P \rho P)$ ? If so, how can one prove that? If not, what would be a counterexample?

A somewhat weaker, but sufficient condition would be that $\langle M v, v \rangle \geq \langle PMPv, v \rangle \quad \forall v \in H$, but I still can't prove it.

Thanks for any help.

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    DANG! You're right, sorry about that. What I said is true only if $S$ reduces $M$.2011-09-27

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M = (9 -2; -2 1) P = (1 0; 0 0 ) rho = (9 2; 2 1) 

is a counterexample, I think.

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    Good, thank you very much!2011-09-28