The concept of spectral dichotomy for linear difference equations (Q1336147)

The authors introduce the notions of general exponent, exponential dichotomy and spectral dichotomy for the linear difference equation \((*)\) \(x_{n + 1} = A_ nx_ n\), where \(x_ n \in \mathbb{R}^ k\) for integers \(n \geq 0\), and discuss the relations between these notions. By using the perturbation theory of linear operators in Banach space, the authors prove that the exponentially stability of \((*)\) implies the corresponding property of the perturbed difference equation \(x_{n+1} = (A_ n + B_ n)x_ n\) under appropriate conditions on \(A_ n\) and \(B_ n\).