WEAKLY ALMOST PERIODIC SEMIGROUPS OF OPERATORS

We address the question as to when a motion or almost-orbit u of a strongly continuous semigroup (S(t))t≥0 of operators in a Banach space X will be weakly almost periodic in the sense of Eberlein. In particular, we show (a) that this is the case in practice exactly when u uniquely decomposes as the sum u = S(.)y + φ of an almost periodic motion S(·)y: ℝ+ → X of (S(t))t≥0 and a function φ: ℝ+ → X that vanishes at infinity in a certain weak sense, and (b) that an almost-orbit u of a uniformly bounded C0-semigroup of linear operators will be weakly almost periodic provided only that u has weakly relatively compact range. Our results on existence and representation are then applied to a qualitative study of asymptotic behavior of solutions to the abstract Cauchy problem in which the focus is on almost periodicity properties and ergodic theorems. © 1990 by Pacific Journal of Mathematics.