Spectral characterizations of stable operator semigroups

Abstract

We introduce the notion of local pseudofunction spectrum $\ddot{I}{ƒ}_{PF}(A)$ for the infinitesimal generator $A$ of a bounded $C_0$-semigroup $\mathcal{T}=(T(t){)}_{t\ge 0}$ on a Banach space $X$ and show it is the right spectral concept to deliver a full characterization of the strong stability of $\mathcal{T}$: $$\forall x\in X:\underset{t\to \infty }{\lim }\Vert T(t)x{\Vert }_X=0⟺\ddot{I}{ƒ}_{PF}(A)=\varnothing .$$ We demonstrate how this yields a quick proof of the well-known Arendt-Batty-Lyubich-VÅ© theorem and establish novel stability results through local range density conditions for semigroups whose local pseudofunction spectra are a null subset of the imaginary axis. We also obtain similar stability characterization theorems for individual orbits and for semi-uniform stability. As an application of our results, we provide spectral characterizations of almost periodic $C_0$-semigroups with countable spectrum. In addition, we prove optimal Tauberian theorems of Katznelson-Tzafriri type and discuss connections with Wiener kernels.

30 pages