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 λ ∈ Sp(T ) such that, for every ε > 0, there is x such that T x − λx < ε but T n x > 1 − ε 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 inf n→∞ 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.