What is the standard definition of "projector", "projection" and "projection operator"? What is the precise meaning of "projector", "projection" and "projection operator"? I always thougth those two terms are synonyms, but I have seen both used in a quantum optics paper where the former is not the same as the latter. There the projector was defined as $(I_S|0)$ and the projection operator as
$\begin{pmatrix}
I_S & 0\\
0 & 0 \\
\end{pmatrix}$
with $I_S$ being the identity for a subspace $S$. So the first is a block-vector and the second a block-matrix? I have never seen such a distinction before and could not find any material about that online. 
 A: Usually, people will use all of these words interchangeably, to mean what your source calls a "projection operator". As for what your source means, note that


*

*a linear transformation is a linear map $T: V \to W$ for vector spaces $V$, $W$

*a linear operator is a linear map $T: V \to V$ for a vector space $V$
We are always working with linear transformations, but sometimes one may specify a transformation is an operator to emphasize that the input and output space are the same.
A projection can be thought of in either way. Suppose that the projection maps vectors in $V$ to a subspace $W$ of $V$. Then we can think of it as a linear transformation $T: V \to W$ where $W$ is regarded as a vector space of its own (what your source calls a projector, represented by a non-square matrix), or we can think of it as a linear operator $T: V \to V$ where the image happens to be $W \subset V$ (which your source calls a projection operator). Making such a distinction might be useful if you want to be very explicit about what the spaces are, but it's not standard.
A: They are the same thing. In quantum mechanics, one usually defines a projection operator as $$\hat{P} = |\psi\rangle\langle\psi|$$
This operator then acts on quantum states (vectors) $|\Psi\rangle$ as
$$\hat{P}|\Psi\rangle = |\psi\rangle\langle\psi|\Psi\rangle = \langle\psi|\Psi\rangle|\psi\rangle$$
This is exactly the same as the projector you defined in matrix form, since we can think of $|\psi\rangle\langle\psi|$ as the diagonal components of a matrix. For example, if $|\Psi\rangle = \alpha|0\rangle + \beta|1\rangle$ and $|\psi\rangle = |0\rangle$, we would find that the projector projects out a particular state $$\hat{P}|\Psi\rangle = \alpha|0\rangle$$
In matrix form this would be exactly the same as what you defined, since now
\begin{equation}\label{key}
\hat{P} = \begin{pmatrix}
1 & 0 \\ 
0 & 0
\end{pmatrix},~~~~~ |\Psi\rangle = \begin{pmatrix}
\alpha \\ 
\beta
\end{pmatrix}
\end{equation}
And now 
$$\hat{P}|\Psi\rangle = \begin{pmatrix}
1 & 0 \\ 
0 & 0
\end{pmatrix}\begin{pmatrix}
\alpha \\ 
\beta
\end{pmatrix} = \begin{pmatrix}
\alpha \\ 
0
\end{pmatrix}$$
