I was reading the paper: Unitary similarity of projectors by Dragomir Z. Dokovic. I do not follow a certain step.
The setting is this: $p$ is a projection (not necessarily orthogonal) in a finite dimensional Hilbert space. We suppose that $p$ and $p^\ast$ (adjoint of $p$) don't have a common eigenvector. Further $a$ is an eigenvector of $pp^\ast$. Then, the author concludes that $a$ is an eigenvector of $p$ with eigenvalue 1 as well. How does one conclude this?
Here is the proof in the paper:
Since $a=\frac1\lambda\cdot p(p^*a)$, we have $a\in {\rm range}(p)$, and as $p$ is a projection, $p(a)=a$.