On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups

We prove that a discrete semigroup of bounded linear operators acting on a complex Banach space is uniformly exponentially stable if and only if, for each , the sequence belongs to . Similar results for periodic discrete evolution families are also stated. 1. Introduction The solutions of the autonomous discrete systems or lead to the idea of discrete semigroups. There are a lot of spectral criteria which characterize different types of stability (or other types of asymptotic behavior) of the solutions of above systems. For further results on asymptotic behavior of semigroups, we refer to [1]. New difficulties appear in the study of the nonautonomous systems, especially because the part of the solution generated by the forced term , that is, , is not a convolution in the classical sense. These difficulties may be passed by using the so-called evolution semigroups. The evolution semigroups were exhaustively studied in [2]. Having in mind the well-known results stated in the continuous case, see for example [2, 3], we can say that this method is a very efficient one. See also [4, 5] for recent developments concerning the semigroups of evolution acting on almost periodic function spaces. Recently, the discrete version of [6] was obtained in [7]. In this note, we study the asymptotic behavior of the discrete semigroups in terms of exact admissibility of the space of almost periodic sequences. In this regard, we develop the theory of discrete evolution semigroups on a special space of bounded sequences. Results of this type in the continuous case may be found in [8] and the references therein. However, by contrast with the continuous case, we did not find in the existent literature papers written in the spirit of the present one referring to the discrete evolution semigroups. These results could be new and useful for people whose area of research is restricted to difference equations. 2. Definitions and Preliminary Results Let be a complex Banach space and the Banach algebra of all linear and bounded operators acting on . The norms in and in will be denoted by . Let be the set of all nonnegative integer numbers. A sequence is said to be almost periodic if for any there exists an integer such that any discrete interval of length contains an integer , such that The integer number is called -translation number of . The set of all almost periodic sequences will be denoted by . For further details about almost periodic functions, we refer to the books [9, 10]. The set？？ of all bounded sequences becomes a Banach space when it is endowed with the “sup” norm denoted