Partial Isometries: Introduction Attention
This question has been modified drastically.
It is done so the answer below is still correct.
It is done so to allow more specialized threads.
Problem
How do I deal with partial isometries?
How come that partial isometries give rise to projections?
And what about the that weird characterization given on wiki?
Reference
For more details see wiki's: Partial Isometries
 A: If $\Omega^{\star}\Omega$ is a projection, then it is an orthogonal projection. It is always true that $\mathcal{N}(\Omega^{\star}\Omega)=\mathcal{N}(\Omega)$ and, hence, $\Omega^{\star}\Omega$ is the orthogonal projection onto $\mathcal{N}(\Omega)^{\perp}=\overline{\mathcal{R}(\Omega^{\star})}$. Therefore,
$$\Omega^{\star}\Omega\Omega^{\star}=\Omega^{\star},\\
   \Omega\Omega^{\star}\Omega=\Omega.
$$
(The second follows by adjoint from the first.)
Likewise, if $\Omega\Omega^{\star}$ is a projection, then the same arguments apply as before where $\Omega$ and $\Omega^{\star}$ are swapped, which is enough because the last two expressions give the same two expressions when $\Omega$ and $\Omega^{\star}$ are swapped.
Note: It is convenient to think of a partial isometry $\Omega$ as a unitary map from one closed subspace $V$ of $\mathcal{H}$ to another closed subspace $W$ of $\mathcal{H}$. Then $\Omega^{\star}$ inverts $\Omega$ on its range $W$ and $\Omega$ inverts $\Omega^{\star}$ on its range $V$. That's why $\Omega^{\star}\Omega$ is the projection onto $V$ and $\Omega\Omega^{\star}$ is the projection onto $W$.
