A condition for asymptotic finite-dimensionality of an operator semigroup

Abstract

Let X be a Banach space and let T: X → X be a power bounded linear operator. Put X0 = {x ∈ X ∣ Tnx → 0}. Assume given a compact set K ⊂ X such that lim infn→∞ρ{Tnx, K} ≤ η < 1 for every x ∈ X, ∥x∥ ≤ 1. If \(\eta < \tfrac{1} {2} \), then codim X0 < ∞. This is true in X reflexive for \(\eta \in [\tfrac{1} {2},1) \), but fails in the general case.