Resumen de On the Exponential Stability of Discrete Semigroups

Akbar Zada, Nisar Ahmad, Ihsan Ullah Khan, Faiz Muhammad Khan

• Let q be a positive integer and let X be a complex Banach space. We denote by Z+ the set of all nonnegative integers. Let Pq (Z+, X) is the set of all X-valued, q-periodic sequences. Then P1(Z+, X) is the set of all X-valued constant sequences.

  When q ≥ 2, we denote by P0 q (Z+, X), the subspace of Pq (Z+, X) consisting of all sequences z(.) with z(0) = 0. Let T be a bounded linear operator acting on X. It is known, that the discrete semigroup generated (from the algebraic point of view) of T , i.e. the operator valued sequence T = (T n), is uniformly exponentially stable (i.e.

  limn→∞ ln T n  n < 0), if and only if for each real number μ and each sequences z(.) in P1(Z+, X) the sequences (yn) given by yn+1 = T (1)yn + eiμ(n+1) z(n + 1), y(0) = 0 is bounded. In this paper we prove a complementary result taking P0 q (Z+, X) with some integer q ≥ 2 instead of P1(Z+, X).