Dichotomy and Boundedness of Solutions for Some Discrete Cauchy Problems Constantin Buşe and Akbar Zada

Abstract

Let us denote by ℤ₊ the set of all nonnegative integer numbers. We prove that a square size matrix A of order m having complex entries is dichotomic (i.e., its spectrum does not intersect the set {z∈ℂ:|z| = 1} if and only if there exists a projection P on ℂ^m which commutes with A, and for each number μ∈ℝ and each vector b∈ℂ^m the solutions of the following two Cauchy problems are bounded:

$$ \left\{ \begin{gathered} x_{n + 1} = Ax_n + e^{i\mu n} Pb, n \in \mathbb{Z}_ + \hfill \\ x_0 = 0 \hfill \\ \end{gathered} \right. $$

and

$$ \left\{ \begin{gathered} y_{n + 1} = A^{ - 1} y_n + e^{i\mu n} (I - P)b, n \in \mathbb{Z}_ + \hfill \\ y_0 = 0. \hfill \\ \end{gathered} \right. $$

The result is also extended to bounded linear operators acting on arbitrary complex Banach spaces.