Spectra based on Bohl exponents and Bohl dichotomy for nonautonomous difference equations

TL;DR

This paper introduces two new spectral notions, Bohl spectrum and Bohl dichotomy spectrum, for nonautonomous difference equations, and explores their relationships with existing spectra like exponential dichotomy spectrum.

Contribution

It defines Bohl spectrum and Bohl dichotomy spectrum, proves their properties, and relates them to classical spectra, providing new tools for analyzing nonautonomous difference equations.

Findings

Bohl dichotomy spectrum is the closure of Bohl spectrum.

Bohl dichotomy spectrum is a subset of exponential dichotomy spectrum.

The spectra of upper triangular systems relate to the spectra of their diagonal entries.

Abstract

For nonautonomous linear difference equations with bounded coefficients on $\mathbb{N}$ which have a bounded inverse, we introduce two different notions of spectra and discuss their relation to the well-known exponential dichotomy spectrum. The first new spectral notion is called Bohl spectrum and is based on an extended notion of the concept of Bohl exponents. The second new spectral notion is called Bohl dichotomy spectrum and is based on a relaxed version of exponential dichotomy called Bohl dichotomy. We prove spectral theorems and show that the Bohl dichotomy spectrum is the closure of the Bohl spectrum and also a subset of the exponential dichotomy spectrum. We discuss the spectra of upper triangular systems and how they relate to the spectra of their diagonal entries. An example illustrates the subtle differences between the different notions of spectra.