Apologies first. I am a physicist and my notations and rigour is probably lousy.
If $P$ is an idempotent operator, $P^2 = P$, $P\neq \mathbb1$ and we have $\forall |\psi\rangle$ the relation,
$P.L |\psi\rangle = L|\psi\rangle$, what conclusions can we draw about $L$, which is a linear operator.
1) $L = P$
Are there anything else? If not how does one prove this? Sorry if this is too trivial.
If $P^2=P$, then the following are equivalent.
$PLx=Lx$ for every $x$.
The range of $L$ is contained in the range of $P$.
If you were talking about a specific vector $x$, then the following are equivalent:
$PLx=Lx$
$Lx$ is in the range of $P$.
Does that answer your question?