On the invertibility of elementary operators

TL;DR

This paper investigates conditions under which elementary operators of length 2 on Banach space operator algebras are invertible or have solutions to specific equations, providing characterizations of invertible cases.

Contribution

It offers necessary and sufficient conditions for solutions to operator equations and characterizes invertible elementary operators of length 2 with elementary inverses.

Findings

Identifies conditions for the existence of solutions to ${ m X} \Phi=0$

Characterizes invertible elementary operators of length 2

Provides criteria for elementary operators to have elementary inverses

Abstract

Let $\mathscr{X}$ be a complex Banach space and $\mathcal{L}(\mathscr{X})$ be the algebra of all bounded linear operators on $\mathscr{X}$. For a given elementary operator $\Phi$ of length $2$ on $\mathcal{L}(\mathscr{X})$, we determine necessary and sufficient conditions for the existence of a solution of the equation ${\rm X} \Phi=0$ in the algebra of all elementary operators on $\mathcal{L}(\mathscr{X})$. Our approach allows us to characterize some invertible elementary operators of length $2$ whose inverses are elementary operators.