Let $\{A_\alpha\}$ be a set of hermitian operators and $\{\Pi(\varepsilon)\}$ a set of projectors on the (finite-dimensional) $\varepsilon$ subspace. Define $$A_\alpha(\Delta\varepsilon)=\sum_{\varepsilon\varepsilon'\atop{\varepsilon-\varepsilon'=\Delta\varepsilon}}\Pi(\varepsilon)A_\alpha\Pi(\varepsilon') $$ (notice how the sum is on every $\varepsilon$ and $\varepsilon'$ such that the difference of the two is $\Delta\varepsilon$). Given $H=\sum_\varepsilon\varepsilon\Pi(\varepsilon)$, I should prove that the $A_\alpha(\Delta\varepsilon)$ are eigenvectors of the superoperator $[H,\cdot]$.
My own calculations don't quite give the correct result... could anyone give a quick proof?