An extension of the Vu–Sine theorem and compact-supercyclicity

If (Tt )t 0 is a bounded C0-semigroup in a Banach space X and there exists a compact subset K ⊆ X such that lim inf t→∞ ρ(Tt x,K) = 0 ∀x ∈ X, x 1 , then there exists a finite-dimensional subspace L ⊆ X such that lim t→∞ ρ(Tt x,L) = 0 (∀x ∈ X). If T :X →X (X is real or complex) is supercyclic and ( T n )n is bounded then (T nx)n vanishes for every x ∈ X. We define the “compact-supercyclicity.” If dimX=∞then X has no compact-supercyclic isometries. © 2006 Elsevier Inc. All rights reserved.