While going through the book on ellipsoidal harmonics, I came across this argument, The spectral form of the ellipsoidal Beltrami operator: $$Be(ρ)S_m^ n (μ, ν) = [(h_2^3 + h_2^2)p_m^n− n(n + 1)ρ^2]S_m^n (μ, ν)$$ on any ellipsoidal surface confocal to the defining reference ellipsoid. For any fixed $\rho$ and any pair of indices $(n,m)$, the quantity $[(h_2^3 + h_2^2)p_m^n− n(n + 1)ρ^2]$ is an eigenvalue of the operator $Be(\rho)$ with corresponding eigenvector the surface harmonic $S_m ^n (μ, ν)$. Hence, in the ellipsoidal case, the eigenspaces of the Beltrami operator are one-dimensional.
I did not understand how it is one dimensional but in the case of spherical harmonics it is $2n+1$ dimensional and what is the consequence of eigenspace being one-dimensional?