Quasi constricted linear representations of abelian semigroups on Banach spaces

2002-07-24
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.
Mathematische Nachrichten

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Citation Formats
E. Emelyanov, “Quasi constricted linear representations of abelian semigroups on Banach spaces,” Mathematische Nachrichten, vol. 233-234, pp. 103–110, 2002, Accessed: 00, 2021. [Online]. Available: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=0036067338&origin=inward.