Exponential Stability of Operators and Operator Semigroups

Extending earlier results of Datko, Pazy, and Littman on C0-semigroups, and of Przyluski and Weiss on operators, we prove the following. Let T be a bounded linear operator on a Banach space X and let r(T) denotes its spectral radius. Let E be a Banach function space over N with the property that limn→∞||χ {0,...,n−1}||E=∞. If for each x ∈ X and x* ∈ X* the map n ↦ 〈x*, Tnx;〉 belongs to E, then r(T) < 1. By applying this to Orlicz spaces E, the following result is obtained. Let T be a bounded linear operator on a Banach space X and let φ: R+ → R+ be a nondecreasing function with φ(t) > 0 for all t > 0. If ∑∞n =0 φ(|〈x*, Tnx;〉 < ∞ for all ||x|| ||x*|| ≤ 1, then r(T) < 1. Assuming a Δ2-condition on φ, a further improvement is obtained. For locally bounded semigroups T = {T(t)} t ≥ 0, we obtain similar results in terms of the maps t ↦ ||T (t) x||.