Slowly changing vectors and the asymptotic finite-dimensionality of an operator semigroup

Abstract

Let X be a Banach space and let T: X → X be a linear power bounded operator. Put X 0 = {x ∈ X | T n x → 0}. We prove that if X 0 ≠ X then there exists γ ∈ Sp(T) such that, for every ɛ > 0, there is x such that ‖Tx − γx‖ < ɛ but ‖T n x‖ > 1 − ɛ for all n. The technique we develop enables us to establish that if X is reflexive and there exists a compactum K ⊂ X such that lim infn→∞ ρ{T n x,K} < α(T) < 1 for every norm-one x∈ X then codim X 0 < ∞. The results hold also for a one-parameter semigroup.