Is it true that for matrices of any sizes (for which the following makes sense), that if $P$ is a symmetric PSD projection matrix,
$$\|APB\|_2 \leq \|AB\|_2?$$
where $\|\cdot\|$ is the operator norm?
Are there any conditions one can put on $A$ and $B$ that make this true?
No. Counterexample: $APB=\pmatrix{1&1\\ 0&0}\pmatrix{1\\ &0}\pmatrix{1&0\\ -1&0}\ne0$, but $AB=0$.