Tauberian theorems and stability of one-parameter semigroups

Abstract

The main result is the following stability theorem: Let T = ( T ( t ) ) t ⩾ 0 \mathcal {T} = {(T(t))_{t \geqslant 0}} be a bounded C 0 {C_0} -semigroup on a reflexive space X X . Denote by A A the generator of T \mathcal {T} and by σ ( A ) \sigma (A) the spectrum of A A . If σ ( A ) ∩ i R \sigma (A) \cap i{\mathbf {R}} is countable and no eigenvalue of A A lies on the imaginary axis, then lim t → ∞ T ( t ) x = 0 {\lim _{t \to \infty }}T(t)x = 0 for all x ∈ X x \in X .