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.