I wanted to understand the following decomposition of a operator relative $A$ to the direct sum $H \oplus H^{\perp}$ that I have seen in literature (in connection with Stinespring theorem) where $P$ is a projector:
$$ A = \begin{bmatrix} PA|_{H} & P^{\perp}A|_{H}\\ PA|_{H^{\perp}} & P^{\perp}A|_{H^{\perp}} \end{bmatrix} $$
I am familiar with the decomposition:
$$ A = \begin{bmatrix} PAP & P^{\perp}AP\\ PAP^{\perp} & P^{\perp}AP^{\perp} \end{bmatrix} $$
for which the calculation is clear.
I wanted to start to understand this using a matrix like
$$ A = \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix} $$
or a right shift operator.
But that means constructing the restriction $A|_{H}$, how would I go about this?
Any help would be greatly appreciated.