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.