  We expound some results about the relationships between the Jacobi operators
with respect to null vectors on a Lorentzian $\mathcal{S}$-manifold $M$ and the
Jacobi operators with respect to particular spacelike unit vectors on $M$. We
study the number of the eigenvalues of such operators in a $\phi$-null Osserman
Lorentzian $\mathcal{S}$-manifold, under suitable assumptions on the dimension
of the manifold. Then, we generalize a curvature characterization, previously
obtained by the first author for Lorentzian $\phi$-null Osserman
$\mathcal{S}$-manifolds with exactly two characteristic vector fields, to the
case of those with an arbitrary number of characteristic vector fields.
