ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXIII, 2 (2004)
p. 197 - 205

Discrete Methods and Exponential Dichotomy of Semigroups
A. L. Sasu

Abstract.  The aim of this paper is to characterize the uniform exponential dichotomy of semigroups of linear operators in terms of the solvability of discrete-time equations over $N$. We give necessary and sufficient conditions for uniform exponential dichotomy of a semigroup on a Banach space $X$ in terms of the admissibility of the pair $(l^\infty(N, X), c_{00}(N,X))$. As an application we deduce that a $C_0$-semigroup is uniformly exponentially stable if and only if the pair $(C_b(R_+, X), C_{00}(R_+, X))$ is admissible for it and a certain subspace is closed and complemented in $X$.
Keywords: Uniform exponential dichotomy, semigroup of linear operators.

AMS Subject classification:  34D05, 34D09.