Quasi Constricted Linear Representations of Abelian Semigroups on Banach Spaces
✍ Scribed by Eduard Yu. Emel'yanov; Manfred P. H. Wolff
- Publisher
- John Wiley and Sons
- Year
- 2002
- Tongue
- English
- Weight
- 174 KB
- Volume
- 233-234
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
Let (X, • ) be a Banach space. We study asymptotically bounded quasi constricted representations of an abelian semigroup IP in L(X), i. e. representations (Tt) t∈IP which satisfy the following conditions: i) lim t→∞ Ttx < ∞ for all x ∈ X. ii) X 0 := {x ∈ X : lim t→∞ Ttx = 0} is closed and has finite codimension. We show that an asymptotically bounded representation (Tt) t∈IP is quasi constricted if and only if it has an attractor A with Hausdorff measure of noncompactness χ • 1 (A) < 1 with respect to some equivalent norm • 1 on X. Moreover we prove that every asymptotically weakly almost periodic quasi constricted representation (Tt) t∈IP is constricted, i.e. there exists a finite dimensional (Tt) t∈IP -invariant subspace Xr such that X := X 0 ⊕ Xr . We apply our results to C 0 -semigroups.