Dichotomies for triangular systems on Hilbert spaces

TL;DR

This paper explores the connection between exponential dichotomies of triangular linear difference systems and their diagonal counterparts on infinite-dimensional Hilbert spaces, extending finite-dimensional results to a broader context.

Contribution

It generalizes the relationship between exponential dichotomies and admissibility properties from finite to infinite-dimensional Hilbert spaces for triangular systems.

Findings

Established conditions for exponential dichotomies in infinite-dimensional systems

Extended finite-dimensional results to Hilbert space settings

Provided new insights into the structure of triangular systems

Abstract

In this article, we study the relationship between the exponential dichotomy properties of a triangular system of linear difference equations and its associated diagonal system on Hilbert spaces. We stress that all previous results in this direction were restricted to the finite-dimensional case. As in the previous work of the first two authors, we rely on the relationship between exponential dichotomies and the so-called admissibility properties. However, this approach requires nontrivial changes when passing from the finite-dimensional to the infinite-dimensional setting.