self-adjoint operators for Pauli-Runge vector

for the Pauli-Runge vector:

$\displaystyle \vec{A} = \frac{1}{2ze^2m}(\vec{L} \times \vec{p} - \vec{p} \times \vec{L}) + \frac{\vec{r}}{r}$

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

im a little confused as to what self-adjoint means. i have read that if $\displaystyle A^{\dagger}=A$ 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 $\displaystyle [A_i, A_j]$?

also, do I have to show it for each of $\displaystyle A_1, A_2$ and $\displaystyle A_3$ or can I just show it for $\displaystyle A_i$ $\displaystyle (i = 1, 2, 3)$?

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