Stability and Almost Periodicity of Trajectories of Periodic Processes

We prove that if the monodromy operator V of a linear periodic process U(t, τ) in a Banach space E is power-bounded, has countable peripheral spectrum, and if its peripheral point spectrum satisfies a certain natural and simple duality condition (which always holds in reflexive spaces), then every positive trajectory u(τ) = U(0, τ) x, τ ≥ 0, x E, is asymptotically almost periodic. If, in particular, the peripheral point spectrum of V* is empty, then every positive trajectory is asymptotically stable. We also obtain results on almost periodicity of complete bounded trajectories, and consider conditions under which nontrivial complete trajectories exist.