Spectral Properties and Stability of One-Parameter Semigroups

Let A be the infinitesimal generator of a strongly continuous semigroup etA of bounded linear operators in a Banach space X with norm • and assume that Y X is also a Banach space with norm • Y which is stronger than the norm • and Y is dense in X. Moreover, suppose that etAY Y for t 0 and etA is also a strongly continuous semigroup in Y with the infinitesimal generator B. We show, when etA is an isometric group, that (a) if λ σ(A), the spectrum of A, is isolated, then λ σp (A), the point spectrum of A; (b) if σ(B) ∩ (iR) is countable, then σ(A) = σ(B) and σp(B) ( σp (A)) is nonempty. As an application of (a) and (b), we show that if etA is uniformly bounded, σ(B) ∩ (iR) is contained in σc(B) and is countable, than limt → ∞etAx = 0 for all x X, where σc(B) denotes the continuous spectrum of B.