**
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.

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Acta Mathematica Universitatis Comenianae

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