I came across the following (in the context of the Hubbard model from condensed matter theory) "projection operators $P_1,P_2$ that project on the subspaces $H_1$ and $H_2$ of a Hilbert space $H$". They ask me to derive "the effective Hamiltonian $P_1H_{eff}P_1$ that acts on $H_1$" considering that $\vert\Psi_1\rangle = P_1\vert\Psi\rangle$ obeys the Schrodinger equation
$$ P_1H_{eff}P_1\vert\Psi_1\rangle=E\vert\Psi_1\rangle $$
My question: how can such an operator $P_1$ make sense? I would think that it was a map $P_1:H\rightarrow H_1$, but that does not make sense because in the Schrodinger equation above we're acting $P_1$ on $\vert\Psi_1\rangle$, an element of $H_1$. To me, it would make sense that our "effective Hamiltonian on $H_1$" was $P_1H_{eff}P_1^{-1}$, where $H_{eff}:H\rightarrow H$. But this is not what they do.
An orthogonal projection $P$ is simply a linear operator $P:H \to H$ such that $P^2 = P$ and $P^* = P$. For an element $\lvert \Psi_1 \rangle \in H_1 \subset H$, we have $P_1 \lvert \Psi_1 \rangle =\lvert \Psi_1 \rangle$. So everything is well defined here.
Note that $P_1^{-1}$ typically isn't well defined, unless $H_1 = H$ in which case $P_1$ would be the identity.