Let $V$ - finite dimension space, $A : V \to V$ positive semidefinite operator, $P$ - orthoprojector ($P^2 = P = P^*$) $R$ - reflection ($R^2 = 1$, $R^* = R$).
Can we say something nontrivial (any inequalities, e.g) about $\operatorname{Tr}[PA]$ or $\operatorname{Tr}[RA]$?
We can indeed say that $$ |\operatorname{Tr}[PA]| \leq \operatorname{Tr}[A] $$ The same cannot be said for $R$. We do have, however, $$ |\operatorname{Tr}[RA]| \leq \|R\| \operatorname{Tr}[A] $$ Where $\|R\|^2$ is the largest eigenvalue of $R^*R$.
Another interesting result in the case of $P$ is that we can say $$ |\operatorname{Tr}[PA]| \leq \sqrt{\operatorname{rank}(P)\operatorname{Tr}[A^*A]} $$ since $\operatorname{Tr}(P) = \operatorname{rank}(P)$. We also have $$ |\operatorname{Tr}[PA]| \leq \operatorname{rank}(P)\|A\| $$ with $\|\cdot\|$ as defined above.