"Consider an operator $A = r - a$, where $r$ is an operator and $a$ is a constant. Consider only those state kets $V_i$ in the Hilbert space such that $AV_i = 0$ ($A$ acting on $V_i$). Define a projection operator $P_A$ that projects an arbitrary ket on to the states $V_i$'s, i.e. the subspace of kets annihilated by $A$.
Then $P_AAP_A = 0$. So $P_ArP_A = aP_A$."
Now further in the literature which includes whatever I've quoted above, the following is stated:
Suppose $H = (1/r^2)p_\theta^2 + \lambda(r - a)$ (where $p_\theta$ is an operator). It follows that
$P_AHP_A = (1/a^2)p_\theta^2$.
Could anyone please elaborate how the last implication comes about? Is it that since $rV_i = aV_i$, $(1/r^2)V_i = (1/a^2)V_i$?