Dichotomy spectra of triangular equations

Christian Pötzsche

doi:10.3934/dcds.2016.36.423

Article Contents

2016, Volume 36, Issue 1: 423-450. Doi: 10.3934/dcds.2016.36.423

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Dichotomy spectra of triangular equations

Christian Pötzsche^1,

1.
Institut für Mathematik, Alpen-Adria Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt

Received: July 31, 2014

Revised: February 28, 2015

Published: May 2017

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Abstract

Without question, the dichotomy spectrum is a central tool in the stability, qualitative and geometric theory of nonautonomous dynamical systems. In this context, when dealing with time-variant linear equations having triangular coefficient matrices, their dichotomy spectrum associated to the whole time axis is not fully determined by the diagonal entries. This is surprising because such a behavior differs from both the half line situation, as well as the classical autonomous and periodic cases. At the same time triangular problems occur in various applications and particularly numerical techniques.
Based on operator-theoretical tools, this paper provides various sufficient and verifiable criteria to obtain a corresponding diagonal significance for finite-dimensional difference equations in the following sense: Spectral and continuity properties of the diagonal elements extend to the whole triangular system.

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Mathematics Subject Classification: Primary: 34D09; Secondary: 37C60, 39A30, 47B37.

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