Let $Y_{\ell m}$ be a real spherical harmonic, and define an operator on functions $f:S^2\to\mathbb{R}$ by $$(L_{\ell m}f)(\theta,\phi) = Y_{\ell m}(\theta,\phi)f(\theta,\phi).$$
What are the eigenfunctions of $L_{\ell m}$ (which must be the same independent of $\ell$ or $m$)? Are there formulas in terms of the spherical harmonics themselves? Are there formulas for the eigenvalues, as functions of $\ell$ and $m$?