Let $n \times n$ matrix $A$ be positive semidefinite and let $P$ be an orthogonal projector of some subspace of $\mathbb{R}^n$ into $\mathbb{R}^n$. Is the following correct?
$$\left( \forall x \in \mathbb{R}^n \right) \left( x^T P A P x \leq x^T A x \right)$$
No. E.g. consider $a\ge1$ and $$ A=\pmatrix{a&1\\ 1&1},\ P=\pmatrix{1&0\\ 0&0},\ x=\pmatrix{1\\ -1}. $$ We have $x^TPAPx=a>a-1=x^TAx$. In fact, $$ A-PAP=\pmatrix{0&1\\ 1&1} $$ is not positive semidefinite.