Dichotomies uniform on subspaces and formulas for dichotomy spectra

TL;DR

This paper introduces a generalized notion of dichotomy that unifies classical exponential and Bohl dichotomies, providing a spectral theorem and formulas for spectral intervals applicable to both spectra.

Contribution

It develops a new unified framework for dichotomies uniform on subspaces and derives spectral formulas, extending previous concepts in the field.

Findings

Established a dichotomy spectral theorem based on the new notion.

Derived formulas for dichotomy spectral intervals for both spectra.

Unified the treatment of exponential and Bohl dichotomies.

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.