The dichotomy spectrum for noninvertible systems of linear difference equations

In this paper we introduce the so-called dichotomy spectrum for nonautonomous linear difference equations x(k+1) = 4(k)x(k), k greater than or equal to kappa (0) greater than or equal to - infinity, whose coefficient matrices A(k) is an element of R-N x N are not supposed to be invertible. This new kind of spectrum is based on the notion of exponential forward dichotomy and consists of at most N+1 closed, not necessarily bounded, intervals of the positive real line. If all the matrices A(k) are invertible then the number of spectral intervals is at most N, and if in addition A(k) equivalent to A is independent of k then these intervals reduce to the absolut values of the eigenvalues of A, In any case the spectral intervals are associated with invariant vector bundles comprising solutions with a common exponential growth rate. The main result of this paper is a Spectral Theorem which describes all possible forms of the dichotomy spectrum.