Exponential dichotomy for noninvertible linear difference equations

Abstract

In this article we study exponential dichotomies for noninvertible linear difference equations in finite dimensions. After giving the definition, we study the extent to which the projection $P\left(k\right)$ in a dichotomy is unique. For equations on $\mathbb{Z}$ it is unique but for equations on ${\mathbb{Z}}_{+}$ only its range is unique and for ${\mathbb{Z}}_{-}$ only its nullspace. Here we strengthen Kalkbrenner's results and give a complete characterization of all possible projections. Next we study the possibility of extending the dichotomy to a larger interval. We reproduce the results of Pötzsche but also show exactly when the original projection remains unchanged. Next we prove that the roughness theorem, well known for additive perturbations, holds for multiplicative perturbations also. The proof uses ideas of Zhou, Lu and Zhang. Finally, following Ducrot, Magal and Seydi, we mention that the results by Palmer on finite time conditions on dichotomy for the invertible case can be extended to the noninvertible case.

Keywords:

• Linear
• difference equations
• noninvertible
• exponential dichotomy
• roughness

2020 Mathematics Subject Classifications:

• 39A06
• 39A30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

K. J. Palmer partially supported by Marche Polytechnic University.