In this paper, the formulas of commutator calculus are applied to the investigation of the stability of linear impulsive differential equations. It is assumed that the moments of impulse action satisfy the average dwell-time (ADT) condition. Sufficient conditions for the asymptotic stability of linear impulsive differential equations in a Banach space are obtained. In the Hilbert space, the stability of the original linear differential equation is reduced to the investigation of a linear differential equation with equidistant moments of impulse action and perturbed discrete dynamics. This reduction simplifies the application of Lyapunov's direct method and the construction of Lyapunov functions. We give examples in the spaces R-2 and X = C[0, l] to illustrate the effectiveness of results obtained. Finally, a sufficient generality of the obtained results on the dynamic properties of linear operators of the linear impulsive differential equation is established. (C) 2018 Elsevier B.V. All rights reserved.