World Applied Sciences Journal 27 (12): 1630-1636, 2013 ISSN 1818-4952 © IDOSI Publications, 2013 DOI: 10.5829/idosi.wasj.2013.27.12.1423 Corresponding Author: Akbar Zada, Department of Mathematics, University of Peshawar, Peshawar, Pakistan 1630 Discrete Characterization of Exponential Dichotomy of Evolution Family over Finite Dimensional Spaces A. Zada, R. Amin, T. Hussain and M. Asif Department of Mathematics University of Peshawar, Peshawar, Pakistan Abstract: The main aim of this article is to discuss the dichotomy of the system u n+1 -A n u n , where (A n ) is a sequence of N-periodic square size matrices of order m with complex entries and U = {U(m,n)} m ≥n≥0 be the N-periodic discrete evolution family generated by bounded linear operators over C m , where N≥ 2. We prove that if the operator U(N,0) is dichotomic, then there exist two positive constants k 1 and k 2 such that: n i (j1) 0 1 j1 U(n,0)Px U(n,j)e Pb k µ − = + ≤ ∑ and n 1 i(j1) 0 2 j1 U(n,0)Qy U (n,j)e Qb k − µ − = + ≤ ∑ We also prove a theorem about the converse of this result. Key words: Evolution family • exponential dichotomy • periodic non-autonomous discrete systems • spectrum • projection • cauchy problem INTRODUCTION Let m ≥ 1 and N≥2 be two natural numbers and let U = {U(p,q)} p≥q≥ 0 be the N-periodic discrete evolution family of square size matrices of order m, with complex entries, generated by bounded linear operators over C m . If the solution of the following discrete Cauchy sequence in n 1 n n 0 u Au e b,n Z ,u 0 µ + + = + ∈ = is bounded for each real number µ and each m-vector b then the operator U(N, 0) is stable. The converse statement is also true if we add a new assumption to the boundedness condition. This new assumption refers to the invertibility of the matrix ( ) N i 1 V U N, e µυ µ υ= = υ ∑ for each real number µ [7] i.e. the operator U(N, 0) is dichotomic if and only if the matrix V µ is invertible and there exists a projection P which commutes with the operator U(N, 0) and the matrix V µ , such that for each real number µ and each vector b ∈C m the solutions of the discrete Cauchy sequences in n 1 n n 0 u Au e Pb,u 0 µ + = + = and 1 in n 1 n n 0 w A w e (I P)b,w 0 − µ + = + − = are bounded [9]. Our paper is the extension of the last coated result [9]. The first section of this article contain some preliminaries, in the second section we give the main results. NOTATIONS AND PRELIMINARY RESULTS In this paper we denote the Banach algebra of all square size matrices of order m with complex entries by X endowed with the usual operatorial norm and the space of all bounded linear operators by B(C m ) An eigenvalue of a matrix A∈X is any complex scalar λ having the property that there exists a nonzero vector v∈C m such that Av = λ v. The spectrum of the matrix A, denoted by σ(A), consists of all its eigenvalues. The resolvent set of A, denoted by ρ (A), is the complement in C of σ (A). Consider 1 {z C:z 1} Γ= ∈ = , 1 {z C:z 1} + Γ = ∈ > and