A discrete Lyapunov theorem for the exponential stability of evolution families

We propose a discrete time approach for the exponential stability of evolution families on a Hilbert space by proposing a Liapunov-type equation which involves only discrete time arguments. The result of A.M. Lyapunov has come into widespread usage in many topics of mathematics. In particular, it continues to be of great importance in modern treatments of the asymptotic behaviour of the solutions of differential systems. Let us recall that the theorem of Lyapunov states that if A is an n× n complex matrix then A has all its characteristic roots with real parts negative if and only if for any positive definite Hermitian H there exists an unique positive definite Hermitian matrix B satisfying the equation A∗B +BA = −H (where ∗ denotes the conjugate transpose of a matrix) (see [1]). This very familiar result was extended in a natural way to strongly continuous semigroups of operators on a complex Hilbert space, by R. Datko [7]. The result of Datko requires the mathematical sophistication of the modern functional analysis tools. A similar result is given by Krein and Daleckij in [6] in the case of the semigroup T (t) = e where A is a bounded linear operator, first for exponential stability and then for exponential dichotomy. Also, in this context results related to the passing from the bounded linear operator A to the case of an unbounded one, can be found in the papers due to C. Chicone [3], J. Goldstein [8], Y. Latushkin [3, 11], S. Montgomery-Smith [11], L. Pandolfi [12], A. Pazy [13] and Vu Quoc Phong [16, 17]. Let B(X) be the Banach algebra of all bounded linear operators acting on the Hilbert space X. The B(X)-valued function T = {T (t)}t≥0 is a semigroup of linear operators if: • T (0) is the identity on X. • T (t+ s) = T (t)T (s) for all t, s ≥ 0. Received May 5, 2005. Mathematics Subject Classification. 34D05, 34D20, 47D06.