Quasi Constricted Linear Representations of Abelian Semigroups on Banach Spaces

Let ( X , || · ||) be a Banach space. We study asymptotically bounded quasi constricted representations of an abelian semigroup ℙ in L ( X ), i.e. representations ( T t ) t ∈ ℙ which satisfy the following conditions: i) $ \overline {lim} _{t \to \infty}||T _{t}x < \infty $ for all x ∈ X . ii) X 0 ≔ { x ∈ X : lim t → ∞ || T t x || = 0} is closed and has finite codimension. We show that an asymptotically bounded representation ( T t ) t ∈ ℙ 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 ( T t ) t ∈ ℙ is constricted, i.e. there exists a finite dimensional T t ) t ∈ ℙ ‐invariant subspace X r such that X ≔ X 0 ⊕ X r . We apply our results to C 0 ‐semigroups.