Discrete (h,k)-Dichotomy and Remarks on the Boundedness of the Projections

Abstract

The present paper treats a concept of (h,k)-dichotomy for linear discrete systems. Sufficient conditions for the k-boundedness of the projection sequences that give the dichotomy are presented and an illustrative example shows the connection between the growth of the system and the bound of the sequence of projections. Thus the growth of the system that is assumed in the theorems is essential.

On (h, k)-Dichotomy of Linear Discrete-Time Systems in Banach Spaces

“…Let (A, P) be a pair which is (h, k) − d. with (P n ) strongly invariant for (A). By Example 3 there exists a sequence of norms N 1 = { | · | n , n ∈ N}, given by (6), compatible with (P n ). We only have to prove the two inequalities.…”

“…We only have to prove the two inequalities. By the fact that (P n ), (Q n ) are orthogonal, replacing x by P n x or Q n x in (6) and since [m, ∞) ⊆ [n, ∞), [0, n] ⊆ [0, m] for all (m, n) ∈ ∆ we obtain that:…”

The Equivalence of Datko and Lyapunov Properties for (h,k)-Trichotomic Linear Discrete-Time Systems

“…In the last few years an important development has been made in the field of the asymptotic behaviors of dynamical systems. Among the most important asymptotic behaviors studied, we mention the properties of stability, dichotomy, and trichotomy (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the references therein).…”

“…An important generalization of the dichotomy concept (approached in various manners in [2,3,6,18]) is the notion of trichotomy, the most complex asymptotic property of dynamical systems. The trichotomy supposes the splitting of the state space, at any moment, into three subspaces: the stable subspace, the unstable subspace, and the central subspace.…”