Additive maps preserving the semi-Fredholm domain in spectrum

Abstract

Let $\mathcal{X}$ be an infinite-dimensional complex Banach space and let $\mathcal{B}\left(\mathcal{X}\right)$ be the algebra of all bounded linear operators. In this article we show that an additive surjective map $\phi$ on $\mathcal{B}\left(\mathcal{X}\right)$ preserving the semi-Fredholm domain in spectrum is an automorphism or an antiautomorphism on $\mathcal{B}\left(\mathcal{X}\right)$.