The theory of spherical harmonics shows that $L^2(B)$ is an $SO(3)$   Hilbert sum of  the form

$$L^2(B)= \bigoplus_{n=1}^\infty H_n, $$

where $H_n\subset L^2(B)$ are finite dimensional $SO(3)$-invariant irreducible subspaces. In fact  $H_n$  are all the irreducible  representations of $SO(3)$ and $H_n\cong H_m$ as $SO(3)$-representations iff $n=m$.  

If $K:L^2(B)\to L^2(B) $ is  $SO(3)$-equivariant, then it  indices $SO(3)$-equivariant maps

$$K_{mn} H_m\to H_n,\;\; H_m\ni h\mapsto P_{H_n}Kh\in H_n,  $$

where $P_{H_n}$ is the orthogonal projection onto $H_n$.  Since $H_n,H_m$ are irreducible representations  we deduce from Schur's Lemma that if $m=n$ then $K_{mn}$ is a multiple of the identity and $K_{nm}=0$ if $m=n$.  This proves that $H_n$ is an invariant subspace of $K$ and   the restriction of $K$ to $H_n$ is a multiple of the identity. In other words, the  functions in $H_n$ are eigenfunctions of $K$.