Exponential Instability and Complete Admissibility for Semigroups in Banach Spaces
نویسنده
چکیده
We associate a discrete-time equation to an exponentially bounded semigroup and we characterize the exponential instability of the semigroup in terms of the complete admissibility of the pair (l∞(N, X), l∞(N, X)). As a consequence, we obtain that in certain conditions a C0-semigroup is exponentially unstable if and only if the pair (Cb(R+, X),Cb(R+, X)) is admissible with respect to an integral equation associated with it. We apply our results at the study of the exponential dichotomy of semigroups.
منابع مشابه
Pairs of Function Spaces and Exponential Dichotomy on the Real Line
We provide a complete diagram of the relation between the admissibility of pairs of Banach function spaces and the exponential dichotomy of evolution families on the real line. We prove that if W ∈ H R and V ∈ T R are two Banach function spaces with the property that either W ∈ W R or V ∈ V R , then the admissibility of the pair W R, X , V R, X implies the existence of the exponential dichotomy...
متن کاملOn 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.
متن کاملOn Two-parameter Dynamical Systems and Applications
In this note some useful properties of strongly continuous two-parameter semigroups of operators are studied, an exponential formula for two-parameter semigroups of operators on Banach spaces is obtained and some applied examples of two-parameter dynamical systems are discussed
متن کاملFavard Spaces and Admissibility for Volterra Systems with Scalar Kernel
We introduce the Favard spaces for resolvent families, extending some well-known theorems for semigroups. Furthermore, we show the relationship between these Favard spaces and the Lp-admissibility of control operators for scalar Volterra linear systems in Banach spaces, extending some results in [22]. Assuming that the kernel a(t) is a creep function which satisfies a(0+) > 0, we prove an analo...
متن کاملOn Sylvester Operator Equations, Complete Trajectories, Regular Admissibility, and Stability of C0-semigroups
We show that the existence of a nontrivial bounded uniformly continuous (BUC) complete trajectory for a C0-semigroup TA(t) generated by an operator A in a Banach space X is equivalent to the existence of a solution Π = δ0 to the homogenous operator equation ΠS|M = AΠ. Here S|M generates the shift C0-group TS(t)|M in a closed translation-invariant subspaceM of BUC(R, X), and δ0 is the point eval...
متن کامل