Lyapunov type equation for discrete exponential trichotomies

For a nonautonomous dynamics obtained by a sequence of linear operators acting on an arbitrary Hilbert space, we give a complete characterization of the notion of a uniform exponential trichotomy in terms of what can be considered to be a discrete version of the Lyapunov equation. We then use this characterization to study the stability of exponential trichotomies under small linear and nonlinear perturbations.