Is $PAP$ invertible if $P$ is a projection operator and $A$ is arbitrary? Let $P \neq I$ be a projection operator in a hilbert space, that is $P^2 = P$. Does there exist an operator $A$ such that $PAP$ is invertible?

$\DeclareMathOperator{\Ima}{Im}$ No, since $\Ima PAP \subseteq \Ima P \subsetneq H$ so $PAP$ cannot be surjective.