The article deals with linear differential equations with periodic impulsive action in a Banach space. The moments of impulsive action are assumed to satisfy the ADT condition. On the basis of a new comparison principle, the study of this equation is reduced to the study of an auxiliary linear impulsive differential equation with periodic sequence of dwell-times. This equation is a perturbation of some linear periodic differential equation. Separately, cases when the phase space is semi-ordered with the help of a solid, normal and unflattened cone, and considered linear impulsive differential equation is positive with respect to the cone are considered. In these cases, a linear Lyapunov function is constructed. The second case is when the phase space is a Hilbert space. Here, the Lyapunov function is constructed in the quadratic form. The sufficient conditions for the asymptotic stability of linear impulsive differential equations are established. Examples of the application of these results are given.