A Krein-Like Theorem for the Linearized Vlasov-Poisson Equation

We consider the linearized Vlasov-Poisson equation in the Banach space with the norm $\|\{f_k\}\|=\sum_k k^2\|f_k\|_{W_{1,1}}$. We perturb the equations by changing the equillibrium solution $f_0$. We prove that that always exists an infinitesimal perturbation of $f_0'$ in the $W_{1,1}$ norm can create an instability at any solution of the equation $f_0'(v)=0$. If we restrict to dynamically accessible perturbations we instead recover a result similar to Krein's theorem for linear finite dimensional Hamiltonian systems.