On the Exponential Stability of Discrete Semigroups

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 P-q (Z(+), X) is the set of all X-valued, q-periodic sequences. Then P-1(Z(+), X) is the set of all X-valued constant sequences. When q >= 2, we denote by P-q(0) (Z(+), X), the subspace of P-q (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(n ->infinity) ln parallel to T-n parallel to/n < 0), if and only if for each real number mu and each sequences z(.) in P-1(Z(+), X) the sequences (y(n)) given by {y(n+1) = T(1)y(n) + e(i mu(n+1))z(n +1), y(0) = 0 is bounded. In this paper we prove a complementary result taking P-q(0) (Z(+), X) with some integer q >= 2 instead of P-1(Z(+), X).