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

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.