The things I'm pretty sure I understand: Let's say I have a single particle hamiltonian $H$ represented by a $2$x$2$ matrix, so it has two eigenstates $|\lambda_1\rangle$ and $|\lambda_2\rangle$. I can define two projectors onto these states $P_1=|\lambda_1\rangle\langle\lambda_1|$ and $P_2=|\lambda_2\rangle\langle\lambda_2|$. Now if we look at a 2-particle system, the Hamiltonian of particle 1 in the direct product space is $H_1=H\otimes I$ and the Hamiltonian of particle 2 is $H_2=I\otimes H$. If I understand correctly, the new space is spanned by, for example, the following 4 states: $|\lambda_i\rangle\otimes|\lambda_j\rangle$, for $i,j=1,2$. $H_1$ will have 4 eigenvectors which correspond to the energy eigenstates $H_1|n\rangle=E_n|n\rangle$ of particle 1 in the two particle system. I can then construct 4 projectors onto these states which are defined in a similar way as above, and I will call them $P'_n$ for $n=1,2,3,4$.
The question: I want to show that, for an arbitrary state in the two-particle vector space, $P'_nf(E_n)|\psi\rangle=P'_nf(H_1)|\psi\rangle$ where $f(E_n)$ is some function of the eigenvalues of the Hamiltonian and $|\psi\rangle$. I know how to do this for a one particle system, but I have no intuition for direct products yet and I can't work out how to reconcile the fact that $H$ has 2 states but $H_1$ has 4 in an attempt to express $P'_n$ in terms of $P_1$ and $P_2$. I can expand in terms of the two particle basis $|\psi\rangle=\sum_{i,j}c_ib_j|\lambda_i\rangle\otimes|\lambda_j\rangle$. Then apply the projector $|n\rangle\langle n|$ but at this point I'm out of ideas. Is there some way to write the eigenfunctions of $H_1$ in terms of the eigenfunctions of $H$?