Weakly almost periodic semigroups of operators
We address the question as to when a motion or almost-orbit u of a strongly continuous semigroup \((S(t))_{t\geq 0}\) 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 \(u=S(\cdot)y+\phi\) of an almost periodic motion S(\(\cdot)y:{\mathbb{R}}\) \(+\to X\) of \((S(t))_{t\geq 0}\) and a function \(\phi\) :\({\mathbb{R}}\) \(+\to X\) that vanishes at infinity in a certain weak sense, and (b) that an almost-orbit u of a uniformly bounded \(C_ 0\)-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.
weakly almost periodic in the sense of Eberlein
uniformly bounded \(C_ 0\)-semigroup of linear operators
qualitative study of asymptotic behavior of solutions to the abstract Cauchy problem
scientific article; zbMATH DE number 4069644