Right (Or Left) Invertibility of Bounded and Unbounded Operators and Applications to the Spectrum of Products

TL;DR

This paper investigates the conditions under which the spectra of the products of two linear operators are equal, focusing on invertibility properties of bounded and unbounded operators and their spectral implications.

Contribution

It establishes the equality of spectra for operator products using invertibility concepts, extending known results to unbounded operators.

Findings

Proves $\sigma(AB)=\sigma(BA)$ for certain classes of operators

Extends spectral equality results to unbounded operators

Provides new insights into invertibility and spectral theory

Abstract

This paper is mainly concerned with proving $\sigma(AB)=\sigma(BA)$ for two linear and non necessarily bounded operators $A$ and $B$. The main tool is left and right invertibility of bounded and unbounded operators.