Viorica Mariela Ungureanu-Exponential stability of stochastic discrete-time, periodic systems in Hilbert spaces 209 EXPONENTIAL STABILITY OF STOCHASTIC DISCRETE-TIME, PERIODIC SYSTEMS IN HILBERT SPACES by Viorica Mariela Ungureanu Abstract. In this paper we consider the linear discrete time systems with periodic coefficients and independent random perturbations (see [4] for the finite dimensional case). We give necessary and sufficient conditions for the exponential stability property of the discussed systems. In order to obtain these characterizations we use either the representations of the solutions of these systems obtained by the authoress in [5] or the Lyapunov equations. These results are the periodic versions of those given in [5]. Key Words: periodic systems, exponential stability, Lyapunov equations. 1. Introduction In this paper we treat the problem of the exponential and uniform exponential stability of time-varying systems described by linear difference equations, with periodic coefficients, in Hilbert spaces. We yield some characterizations of the uniform exponential stability property, which used the two representation theorems of the solutions of these systems given in [5]. We also prove that in the periodic case (but not in the general case), the uniform exponential stability is equivalent with the exponential stability. Another necessary and sufficient conditions for the exponential stability was obtained in terms of Lyapunov equations. 2.Preliminaries Let H be a real separable Hilbert space and L(H) be the Banach space of all bounded linear operators transforming H into H. We write 〈. , .〉 for the inner product and ║.║ for norms of elements and operators. We denote by a ⊗ b, a, b ∈ H the bounded linear operator of L(H) given by a ⊗ b(h) = 〈h, b〉 a for all h ∈ H. 2.1 Nuclear operators The operator A ∈ L(H) is said to be nonnegative, and we write A ≥ 0, if A is self adjoint and 〈Ax, x〉 ≥ 0 for all x ∈ H. We say that A ∈ L(H) is a positive operator (A > 0) if there exists γ > 0 such that A > γI, where I is the identity operator on H. For A ∈ L(H), A ≥ 0 we denote by A ½ the square root of A (see [2]) and by |A| the operator (A*A) ½ . Let A ∈ L(H), A ≥ 0 and{e n } n ∈ N* be an orthonormal basis in H. We define