Eigenvalues of rotation invariant operators on 2-sphere

Question

Work on $L^2(S^2)$, where $S^2$ is the 2-sphere. Suppose that I have an operator, $T$, that is rotation invariant. That is, $T$ commutes with $R$ for any rotation operator $R$. Suppose furthermore that $T$ is compact. I am under the impression that $T$ can be diagonalized using spherical harmonics? Moreover, if $T$ is given by given an integral kernel $V \in L^1(R^3)$, that is, $$ (Tf)(x) = \int \hat{V}(p-q)f(q) dw(q) $$ where $dw$ is the usual measure on $S^2$ and $\hat{V}$ is the Fourier transform of $V$. Then the eigenvalues are, up to factors of $2\pi$, $$ \int V(x) |j_l(|x|)|^2 dx $$ where $j_l$ denotes the spherical Bessel functions. Is this true? how do you prove this?