# Further common local spectral properties for bounded linear operators

## TL;DR

This paper investigates common local spectral properties of certain bounded linear operators, establishing shared spectral features and properties under specific algebraic conditions, with applications to Fredholm operators.

## Contribution

It introduces new algebraic conditions under which bounded linear operators share various local spectral properties and explores their implications.

## Key findings

- AC and BA share the single valued extension property
- They also share the Bishop property (β) and related spectral properties
- Applications to Fredholm operators are provided

## Abstract

In this note, we study common local spectral properties for bounded linear operators $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $$A(BA)^2=ABACA=ACABA=(AC)^2A.$$ We prove that $AC$ and $BA$ share the single valued extension property, the Bishop property $(\beta)$, the property $(\beta_{\epsilon})$, the decomposition property $(\delta)$ and decomposability. Closedness of analytic core and quasinilpotent part are also investigated. Some applications to Fredholm operators are given.