A (μ,ν)-dichotomy spectrum

Abstract

In this work, we introduce three notions of dichotomy spectrum based on general growth rates and describe their structure. Our results are applicable to nonautonomous linear systems acting on general Banach spaces having negative μ-index of compactness, a condition which is satisfied, for instance, by any sequence of compact operators. Moreover, for any possible form of the spectra, we present an explicit example exhibiting such spectrum. Furthermore, as an application, we obtain normal forms of certain nonautonomous systems. We emphasize that the classical Sacker-Sell spectrum can be obtained as a very particular case of our setting.