Property $(UW_E)$ and Consistent invertibility

TL;DR

This paper investigates the stability of Property $(UW_E)$ under various perturbations and operations, focusing on spectral properties and invertibility in Hilbert space operators.

Contribution

It introduces new results on the stability of Property $(UW_E)$ under commuting finite rank perturbations, direct sums, and tensor products.

Findings

Property $(UW_E)$ is stable under commuting finite rank perturbations.

Stability of Property $(UW_E)$ under direct sum and tensor product operations.

Analysis of spectrum related to consistent invertibility and upper semi-Weyl spectrum.

Abstract

In this paper, we study the stability of Property $(UW_E)$ under commuting finite rank perturbation in terms of spectrum induced by consistent in invertibility for operators and a variant of upper semi-Weyl spectrum. Also we discuss the stability of Property $(UW_{E})$ under direct sum $T\oplus S$ and tensor product $T\otimes S$ for Hilbert space operators $T$ and $S$.