# A note on the common spectral properties for bounded linear operators

## TL;DR

This paper investigates spectral properties of bounded linear operators on Banach spaces, establishing conditions under which their spectra, excluding zero, are equal for certain operator products.

## Contribution

It introduces new spectral equivalence results for operator products under specific algebraic conditions, expanding understanding of spectral behavior in Banach space operators.

## Key findings

- Spectral equality for operator products under given conditions
- Extension of spectral theory for regularities in Banach spaces
- Conditions ensuring spectral invariance excluding zero

## Abstract

Let $X$ and $Y$ be Banach spaces, $A\,:\,X\rightarrow Y$ and $B,\,C\,:\,Y\rightarrow X$ be bounded linear operators. We prove that if $A(BA)^2=ABACA=ACABA=(AC)^2A,$ then $$\sigma_{*}(AC)\setminus\{0\}=\sigma_{*}(BA)\setminus\{0\}$$ where $\sigma_*$ runs over a large of spectra originated by regularities.