van Neerven, J. M. A. M.; Priola, E. Norm discontinuity and spectral properties of Ornstein–Uhlenbeck semigroups. (English) Zbl 1117.47035 J. Evol. Equ. 5, No. 4, 557-576 (2005). Let \(BUC(E)\) the space of bounded real-valued uniformly continuous functions on a real Banach space \(E\). The authors consider an Ornstein–Uhlenbeck semigroup \(P=\{P(t)\}_{t\geq 0}\) defined on \(BUC(E)\) such that \(P\) admits an invariant measure. They investigate the condition \[ \|P(t)-P(s)\| = 2, \quad s,t\geq 0,\;s\neq t, \tag{*} \] where the norm is taken in \({\mathcal L}(BUC(E))\), the Banach space of all bounded linear operators on \(BUC(E)\). They prove that \((*)\) holds, under some regularity assumptions. Moreover, the behaviour of \(P\) in \(BUC(E)\) and a dichotomy related to \((*)\) are studied. Some results of the authors are new even for finite-dimensional spaces. As an application, they investigate the spectrum of the generator associated to \(P\). Reviewer: Mohamed Hmissi (Tunis) Cited in 3 Documents MSC: 47D07 Markov semigroups and applications to diffusion processes 35J70 Degenerate elliptic equations 35P05 General topics in linear spectral theory for PDEs 35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) 60J35 Transition functions, generators and resolvents Keywords:Ornstein-Uhlenbeck semigroup; norm discontinuity; spectrum; invariant measure PDFBibTeX XMLCite \textit{J. M. A. M. van Neerven} and \textit{E. Priola}, J. Evol. Equ. 5, No. 4, 557--576 (2005; Zbl 1117.47035) Full Text: DOI arXiv