Abstract
Abstract For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. As an application of the spectral theorem, we prove a reducibility result.
Highlights
Let Ak ∈ RN×N, k ∈ Z, be a sequence of invertible matrices
For nonautonomous linear di erence equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem
As an application of the spectral theorem, we prove a reducibility result
Summary
Let Ak ∈ RN×N , k ∈ Z, be a sequence of invertible matrices. We consider the following nonautonomous linear di erence equations xk+ = Akxk,. Let Φ : Z × Z → RN×N , (k, l) → Φ(k, l), denote the evolution operator of (1.1), i.e., A k−. · · · Al, for k > l, Φ(k, l) = Id, for k = l, A−k · · · A−l− , for k < l. An invariant projector of (1.1) is de ned to be a function P : Z → RN×N of projections Pk , k ∈ Z, such that for each Pk the following property holds. We say that (1.1) admits an exponential dichotomy if there exist an invariant projector P and constants
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