Evolutionary Semigroups and Lyapunov Theorems in Banach Spaces

TL;DR

This paper develops spectral and hyperbolicity criteria for operator semigroups in Banach spaces, linking the properties of the generator to the behavior of solutions of nonautonomous differential equations.

Contribution

It introduces a spectral mapping theorem for continuous semigroups and characterizes hyperbolicity via the generator of an evolutionary semigroup in Banach spaces.

Findings

Spectral mapping theorem for semigroups in Banach spaces

Hyperbolicity condition in terms of generator of evolutionary semigroup

Discrete technique for analyzing evolutionary semigroups

Abstract

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 the investigating of the evolutionary semigroup is developed and applied to describe the hyperbolicity (exponential dichotomy) of the nonautonomuos equation. File Length: 68K