Generalized Evolution Semigroups and $h-$Dichotomies for Evolution Families on Banach Spaces
Álvaro Castañeda, Verónica Poblete, Gonzalo Robledo
TL;DR
The paper develops a generalized dichotomy theory for evolution families on Banach spaces by introducing an $h$-time framework, where decay/growth is governed by a strictly increasing homeomorphism $h$. It constructs $h$-evolution families and associated $h$-semigroups on $C_0(\mathbb{R}_*,X)$, and uses a conjugacy to classical semigroups to translate spectral information. The main contributions are (i) an algebraic/topological setup based on the abelian group $(\mathbb{R},*_h)$ and the vector space $(\mathbb{R},*_h,\odot)$ with a complete norm $|\cdot|_*$, (ii) a precise generator $B_h$ for the $h$-semigroup and its resolvent relationship to the classical generator $A$, and (iii) a full spectral characterization: $U$ has an $h$-dichotomy if and only if $\{T_{t_0}\}\ $ is $h$-hyperbolic and $\sigma(B_h)\cap i\mathbb{R}=\varnothing$, with an explicit formula for $B_h^{-1}$. These results extend classical exponential dichotomy theory to a broad class of nonexponential decays and provide practical tools for analyzing long-time behavior under general time-scaling.
Abstract
This paper develops a comprehensive theory generalizing exponential decay patterns for evolution processes in Banach spaces. We replace classical exponential bounds with more flexible decay rates governed by an increasing homeomorphism $h$. The core of our approach lies in constructing particular group structures induced by $h$, which allow us to define generalized semigroups on function spaces. We prove that these $h$-semigroups are equivalent to classical evolution semigroups through a natural transformation. Our main result establishes that three fundamental concepts are equivalent: hyperbolicity of the generalized semigroup, dichotomy of the underlying evolution process, and a spectral condition on the generator. This work extends classical dichotomy theory to encompass a wider class of decay patterns, providing new tools for analyzing asymptotic behavior in dynamical systems.