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
