Latushkin, Yuri; Montgomery-Smith, Stephen Evolutionary semigroups and Lyapunov theorems in Banach spaces. (English) Zbl 0878.47024 J. Funct. Anal. 127, No. 1, 173-197 (1995). Summary: We present a spectral mapping theorem for continuous semigroups of operators on any Banach space \(E\). The condition for the hyperbolicity of a semigroup on \(E\) is given in terms of the generator of an evolutionary semigroup acting in the space of \(E\)-valued functions. The evolutionary semigroup generated by the propagator of a nonautonomous differential equation in \(E\) is also studied. A “discrete” technique for investigating the evolutionary semigroup is developed and applied in describing the hyperbolicity (exponential dichotomy) of the nonautonomous equation. Cited in 3 ReviewsCited in 40 Documents MSC: 47D06 One-parameter semigroups and linear evolution equations Keywords:discrete technique; exponential dichotomy; spectral mapping theorem; continuous semigroups of operators; hyperbolicity; propagator of a nonautonomous differential equation; nonautonomous equation PDFBibTeX XMLCite \textit{Y. Latushkin} and \textit{S. Montgomery-Smith}, J. Funct. Anal. 127, No. 1, 173--197 (1995; Zbl 0878.47024) Full Text: DOI arXiv Link