Abstract :
In this paper we investigate the strong asymptotic stability of linear dynamical systems in Banach spaces. Let be the infinitesimal generator of a C0-semigroup et of bounded linear operators in a Banach space X. We first show that if et is a C0-isometric group, then there exists at least one pure imaginary λ = iβ σ( ), the spectrum of , and if et is only a C0-isometric semigroup, but not a group, then λ σr( ), the residual spectrum of , for all λ C with Re λ < 0. Next, as an application of the above, we show that if et is uniformly bounded and Re λ < 0 for all λ σ( ), then et is strongly asymptotically stable, i.e., et x → 0 as t → ∞ for all x X; conversely, if et is strongly asymptotically stable, then it is uniformly bounded and Re λ ≤ 0 for all λ σ( ) and any pure imaginary can be only a continuous spectral point of . Finally, we consider the C0-semigroup et B associated with linear elastic systems with damping + B + Aw = 0 in a Hilbert space H, where B is the closure of B = ([formula]). A very general result with regard to strong asymptotic stability of the semigroup et B is obtained.
