Balanced Exponential Growth of Operator Semigroups

Exponential and Polynomial Dichotomies of Operator Semigroups on Banach Spaces

Let A generate a C0–semigroup T (·) on a Banach space X such that the resolvent R(iτ, A) exists and is uniformly bounded for τ ∈ R. We show that there exists a closed, possibly unbounded projection P on X commuting with T (t). Moreover, T (t)x decays exponentially as t → ∞ for x in the range of P and T (t)x exists and decays exponentially as t→ −∞ for x in kernel of P . The domain of P depends ...

The Discrete Fragmentation Equation: Semigroups, Compactness and Asynchronous Exponential Growth

In this paper we present a class of fragmentation semigroups which are compact in a scale of spaces defined in terms of finite higher moments. We use this compactness result to analyse the long time behaviour of such semigroups and, in particular, to prove that they have the asynchronous growth property. We note that, despite compactness, this growth property is not automatic as the fragmentati...

Connexion with Operator Semigroups

On Exponential Dichotomy of Semigroups

The aim of this paper is to analyze the connections between the exponential dichotomy of a semigroup on a Banach space X and the admissibility of the pair (`p(N, X), `q(N, X)). We obtain necessary and sufficient conditions for exponential dichotomy of exponentially bounded semigroups using discrete time techniques.

Exponential Stability and Exponential Dichotomy of Semigroups of Linear Operators

The aim of this paper is to establish necessary and sufficient conditions for exponential stability of semigroups of linear operators and to show how this conditions can be applied in order to characterize the exponential dichotomy. First, we prove that an exponentially bounded semigroup is exponentially stable if and only if it is (l(N, X), l∞(N, X))-stable, where p ∈ (1,∞). After that this re...