Dichotomies with No Invariant Unstable Manifolds for Autonomous Equations

We analyze the existence of (no past) exponential dichotomies for a well-posed autonomous differential equation (that generates a C0-semigroup {()}≥0). The novelty of our approach consists in the fact that we do not assume the T(t)-invariance of the unstable manifolds. Roughly speaking, we prove that if the solution of the corresponding inhomogeneous difference equation belongs to any sequence space (on which the right shift is an isometry) for every inhomogeneity from the same class of sequence spaces, then the continuous-time solutions of the autonomous homogeneous differential equation will exhibit a (no past) exponential dichotomic behavior. This approach has many advantages among which we emphasize on the facts that the aforementioned condition is very general (since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical ？ spaces, sequence Orlicz spaces, etc.) and that from discrete-time conditions we get information about the continuous-time behavior of the solutions.