Weakly almost periodic semigroups of operators
Wolfgang M. Ruess, William H. Summers · Pacific Journal of Mathematics · 1990
We address the question as to when a motion or almost-orbit u of a strongly continuous semigroup (S(t)) t >o 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 >o 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 Co-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.