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This is a Quantitative Study study.

September 27, 2025

Invertible perturbations of essential spectra for upper triangular relation matrices

XWXiufeng WuInner Mongolia Normal University JHJunjie HuangFoshan University ACAlatancang ChenInner Mongolia Normal University

Key Points

The essential spectrum can differ between bounded and invertible perturbations, highlighting a crucial distinction.
A necessary and sufficient condition for Fredholm relation completion is provided based on bounded invertible relations.
The study offers insights into how upper triangular relation matrices operate under various perturbations of their essential spectra.
The findings can have broader implications for understanding spectra behavior in similar matrix structures.

Abstract

This paper is concerned with the invertible perturbations of the essential spectra for the upper triangular partial relation matrix with given relations. A necessary and sufficient condition is given under which such a relation matrix admits a Fredholm (left Fredholm, or right Fredholm) relation completion by choosing some bounded invertible relation. It is shown that the essential (left essential, or right essential) spectrum under bounded perturbations can be a proper subset of that under invertible perturbations.

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