for the Pauli-Runge vector:
\vec{A} = \frac{1}{2ze^2m}(\vec{L} \times \vec{p} - \vec{p} \times \vec{L}) + \frac{\vec{r}}{r}

where \vec{A} = (A_1, A_2, A_3) denotes the Pauli-Runge operators, show that A_1, A_2 and A_3 are self-adjoint operators

im a little confused as to what self-adjoint means. i have read that if A^{\dagger}=A then A is self-adjoint, but im not sure how to show that for this particular vector A! also, does it have anything to do with the commutation relations [A_i, A_j]?
also, do I have to show it for each of A_1, A_2 and A_3 or can I just show it for A_i (i = 1, 2, 3)?

i then have to show that A_j L_j = 0 = L_j A_j where \vec{L} = (L_1, L_2, L_3) denotes the orbital angular momentum not sure how to do this either!!!