Stability properties characterizing the spectra of operators on Banach spaces (Q1908087)

Given a strongly continuous bounded representation \(T\) on a Banach space \(E\) of a locally compact abelian semigroup \(S\), the spectrum \(\sigma (T)\) of \(T\) is characterized through compactness of the orbit of \(T\) in \({\mathcal L} (E)\). In particular the peripheral spectrum of \(T\) consists only of simple poles if and only if any orbit is asymptotically compact in \({\mathcal L} (E)\). -- These results extend the Katznelson-Tzafriri theorem. Applications to the cases \(S = \mathbb{R}_+\) or \(S = \mathbb{Z}_+\) are given.