+The contributions of this operator act differently on $\ket{l,m}$ and --- in fact --- depend on the respectively considered spinor component, which is incorporated by $\ket{l,m,\pm}$.
+\begin{enumerate}
+\item \underline{$L_+S_-$}:
+ Updates spin down component and only acts on spin up component
+\begin{equation}
+L_+S_-\ket{l,m,+}=L_+\ket{l,m}S_-\ket{+}=
+\sqrt{(l-m)(l+m+1)}\hbar\ket{l,m+1}\hbar\ket{-}
+\end{equation}
+ Moreover, this part only acts on magnetic quantum numbers
+ $m=-l,\ldots,l-1$ and updates quantum numbers $m=-l+1,\ldots,l$.
+\item \underline{$L_-S_+$}:
+ Updates spin up component and only acts on spin down component
+\begin{equation}
+L_-S_+\ket{l,m,-}=L_+\ket{l,m}S_+\ket{-}=
+\sqrt{(l+m)(l-m+1)}\hbar\ket{l,m-1}\hbar\ket{+}
+\end{equation}
+ Moreover, this part only acts on magnetic quantum numbers
+ $m=-l+1,\ldots,l$ and updates quantum numbers $m=-l,\ldots,l-1$.
+\item \underline{$L_zS_z$}: Acts on both and updates both spinor components
+\begin{equation}
+L_zS_z\ket{l,m,\pm}=L_z\ket{l,m}S_z\ket{\pm}=
+\pm\frac{1}{2}m\hbar^2\ket{l,m,\pm}
+\end{equation}
+ It acts on all magnetic quantum numbers and updates all of them.
+\end{enumerate}
+Please note that the $\ket{l,m,\pm}$ are not eigenfunctions of the two combinations of ladder operators, i.e.\ the $\ket{l,m,\pm}$ do not diagonalize the spin-orbit part of the Hamiltonian.