Dichotomies uniform on subspaces and formulas for dichotomy spectra

Abstract

In this note we introduce a notion of dichotomy which generalizes the classical concept of exponential dichotomy and the recent notion of Bohl dichotomy. A key attribute is the discussion of the sets of subspaces of the state space on which the dichotomy estimates are uniform. Two main results are a dichotomy spectral theorem based on our notion of dichotomy which is uniform on subspaces and a formula for the dichotomy spectral intervals which is new for the Bohl dichotomy spectrum as well as for the classical exponential dichotomy spectrum.