×

Evolutionary semigroups and Lyapunov theorems in Banach spaces. (English) Zbl 0878.47024

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.

MSC:

47D06 One-parameter semigroups and linear evolution equations
PDFBibTeX XMLCite
Full Text: DOI arXiv Link