Strong Asymptotic Stability of Linear Dynamical Systems in Banach Spaces

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.