On the Spectrum, Complete Trajectories, and Asymptotic Stability of Linear Semi-dynamical Systems

Let {T(t)}t≥0 be a C0-semigroup in a Banach space X with generator A. We prove that if {T(t)}t≥0 is bounded and sun-reflexive, and the sun-dual semigroup is not asymptotically stable, then there exist bounded complete trajectories under {T(t)}t≥0, provided: (i) there is t0 > 0 such that ran(T (t0)) is dense in X , or (ii) σ(A) i . Questions of almost periodicity of complete trajectories are also discussed and a new proof of our earlier theorem (jointly with Yu. I. Lyubich) on asymptotic stability is given.