A criterion for the exponential stability of linear difference equations

Abstract We give an affirmative answer to a question formulated by Aulbach and Van Minh by showing that the linear difference equation x n +1 = A n x n , for n ∈ N in a Banach space B is exponentially stable if and only if for every f = {fn}n=1∞ ∈ lp( N B), where I x n +1 = A n x n + f n , for n ∈ N , x 1 = 0 is bounded on N .