When reading the paper 


E. Carlen, J. Geronimo & M. Loss: SIAM J. MATH. ANAL., vol. 40, no. 1, 327-374


I found an argument like the following. 


Given an bounded and self-adjoint linear operator $K: L^2(B) \rightarrow L^2(B)$, where $B$ is the unit ball in $\mathbb R^3$ equipped with the Lebesgue measure. Suppose that $K$ commutes with all rotations in $\mathbb R^3$: if $R$ is a rotation on $\mathbb R^3$, then 
$$K(g \circ R) = (Kg) \circ R. $$
Then $K$ has a complete basis of eigenfunctions of the form 
$$g(v) = h(|v|)Y\_l^m(v/|v|), $$
where $Y\_l^m$ is some spherical harmonic function and $h$ is some function on $\mathbb R\_+$. 


It seems to be some well-known fact about the spherical harmonics, but still confused me a lot. I consider in the following way. 


Since $K$ commutes with all rotations $R$, if $\lambda$ is an eigenvalue of $K$ and $X\_\lambda$ is the corresponding eigen subspace, then it is clear that for all $g \in X\_\lambda$, 
$$K(g \circ R) = (Kg) \circ R = \lambda g \circ R. $$
Thus $X\_\lambda$ should be a subspace of $L^2(B)$, which is invariant under all rotations $R$. It is also clear that for fixed $l$, the linear span in $L^2(B)$ of the functions 
$$g(v) = h(|v|)Y\_l^m(v/|v|), \ -l \leq m \leq l $$
is invariant under all $R$. But I am not sure how to continue. 


Could anyone give me a proof or some reference? Thanks in advance. 



The eigenfunctions of an operator commuting with all rotations.
In the paper you mention at page 345 it is shown that the operator $K$ is selfadjoint and, for every $N\geq 0$, the operator $K$ preserves the space $\newcommand{\eP}{\mathscr{P}}$ $\eP\_N$ of polynomials of degree $\leq N$. This space has an $SO(3)$-invariant orthogonal decomposition


$$\eP\_N= \bigoplus\_{k+2\ell\leq N} r^{2\ell} H\_\ell $$


where $H\_k$ denotes the space of degree $k$ harmonic homogeneous polynomial. The spaces $H\_k$ are irreducible $SO(3)$-representations and Schur Lemma will imply that, for every $k$ and $m$ the spaces


$$\bigoplus\_{k=0}^m r^{2\ell} H\_k $$


are $K$-invariant since $K$ is $SO(3)$-equivariant.


