Common eigenfunctions of 0 eigenvalue for the angular momentum components

Question

I need to demonstrate (considering the natural representation in $L^2(S^2)$) that if two non zero vectors in $H^2(S^2)$ satisfy $J_a\phi=0$$ \forall a \in {1,2,3}$ (where $J_a$ are the components of the angular momentum operator https://en.wikipedia.org/wiki/Angular_momentum_operator) then they are equal less than normalization and phase.

Hope I posed my problem comprehensibly, otherwise I apologize. Thank you.

Answer 1

For each one of your vectors, you immediately see that they are in the kernel of the quadratic Casimir of SO(3), so $\sum_a J_a J_a$. That is, they are both SO(3) singlets, and do not vary with angle on a sphere. They are constants. Scalars. You don't even need a similarity transformation to connect them. They are merely proportional.