  We study the linear Vlasov equation with a given electric field $E \in
\mathcal{S}$, where $\mathcal{S}$ is the space of Schwartz functions. The
associated damped partial differential equation has a unique tempered solution,
which fixes the needed Cauchy datum. This tempered solution then converges to
the causal solution of the linear Vlasov equation when the damping parameter
goes to zero. This result allows us to define the plasma conductivity operator
$\sigma$, which gives the current density $j = \sigma (E)$ induced by the
electric field $E$. We prove that $\sigma$ is continuous from $\mathcal{S}$ to
its dual $\mathcal{S}^\prime$. We can treat rigorously the case of uniform
non-magnetized non-relativistic plasma (linear Landau damping) and the case of
uniform magnetized relativistic plasma (cyclotron damping). In both cases, we
demonstrate that the main part of the conductivity operator is a
pseudo-differential operator and we give its expression rigorously. This
matches the formal results widely used in the theoretical physics community.
