  We use a spin-rotational invariant Gutzwiller energy functional to compute
random-phase-approximation-like (RPA) fluctuations on top of the Gutzwiller
approximation (GA). The method can be viewed as an extension of the previously
developed GA+RPA approach for the charge sector [G. Seibold and J. Lorenzana,
Phys. Rev. Lett. {\bf 86}, 2605 (2001)] with respect to the inclusion of the
magnetic excitations. Unlike the charge case, no assumptions about the time
evolution of the double occupancy are needed in this case. Interestingly, in a
spin-rotational invariant system, we find the correct degeneracy between
triplet excitations, showing the consistency of both computations. Since no
restrictions are imposed on the symmetry of the underlying saddle-point
solution, our approach is suitable for the evaluation of the magnetic
susceptibility and dynamical structure factor in strongly correlated
inhomogeneous systems. We present a detailed study of the quality of our
approach by comparing with exact diagonalization results and show its much
higher accuracy compared to the conventional Hartree-Fock+RPA theory. In
infinite dimensions, where the GA becomes exact for the Gutzwiller variational
energy, we evaluate ferromagnetic and antiferromagnetic instabilities from the
transverse magnetic susceptibility. The resulting phase diagram is in complete
agreement with previous variational computations.
