Quasi constricted linear representations of abelian semigroups on Banach spaces


Emel'yanov E. , Wolff M. P. H.

Mathematische Nachrichten, vol.233-234, pp.103-110, 2002 (Journal Indexed in SCI Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 233-234
  • Publication Date: 2002
  • Title of Journal : Mathematische Nachrichten
  • Page Numbers: pp.103-110
  • Keywords: C0-semigroups, Constricted semigroups, Quasi-constricted semi-groups, Semigroups of operators, Weakly almost periodic semigroups

Abstract

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) limt→∞ ∥Ttx∥ < ∞ for all x ∈ X. ii) X0:= {x ∈ X:limt→∞ ∥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 X∥·∥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:= X0 ⊕ Xr. We apply our results to C0-semigroups.