Spectral characterizations of stable operator semigroups

Abstract

We introduce the notion of local pseudofunction spectrum ${\sigma }_{PF}(A)$ 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 $\mathcal{T}$: \[ \forall x \in X : ~ \lim_{t \to \infty} \| T(t) x \|_X = 0 \quad \Longleftrightarrow \quad σ_{PF}(A) = \varnothing. \] We demonstrate how this yields a quick proof of the well-known Arendt-Batty-Lyubich-Vũ theorem and establish novel stability results for semigroups whose local pseudofunction spectra are contained in a null subset of the imaginary axis through localization of range density conditions to the potential null singular set. 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 asymptotically almost periodic $C_0$-semigroups with countable local pseudofunction spectrum. In addition, we prove optimal Tauberian theorems of Katznelson-Tzafriri type and discuss connections with Wiener kernels.

arXiv comment37 pages. The proof of Theorem 8.4 has been corrected; a new section has been added with the transport equation as example