Dichotomy spectrum for difference equations in Banach spaces

Abstract

In this paper, we study the structure and properties of the dichotomy spectrum for linear difference equations in Banach spaces avoiding the commonly made assumption of having a compact flow. In this situation, any compact subset of the positive-half line may occur as a dichotomy spectrum. To overcome this difficulty, we introduce the upper and lower dichotomy index by means of a measure of noncompactness. It allows to decompose the dichotomy spectrum into the upper, essential and lower dichotomy spectrum. The main result is a Spectral Theorem which describes all possible cases of the upper and lower dichotomy spectrum and yields a “nonautonomous linear algebra” appropriate also for systems in the absence of Lyapunov regularity. Furthermore, we give explicit examples of difference equations to illustrate the different cases of the Spectral Theorem and to underline that a new form of the spectrum not present in the literature can occur. Finally, we indicate applications to nonlinear difference equations and the continuous time situation.