Is $PAP$ invertible if $P$ is a projection operator and $A$ is arbitrary?

Question

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?

score 2 · Accepted Answer · answered Sep 18 '17 at 08:08

2

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

answered Sep 18 '17 at 08:08

mechanodroid

score 1 · Answer 2 · answered Sep 18 '17 at 08:09

1

If $P\equiv 0$ then for any $A$ we have $PAP\equiv 0$ hence such $A$ need not tu exist.

answered Sep 18 '17 at 08:09

2 Answers2