Surjective maps preserving quasi-nilpotence of products of operators

Abstract

Let $X$ be an infinite-dimensional complex Banach space and let $\mathcal B(X)$ the full algebra of all bounded linear operator on $X$. In this paper, we give a complete characterization of surjective maps $ϕ:\mathcal B(X)\to\mathcal B(X)$ satisfying \[ r(AB)=0\quad\Longleftrightarrow\quad r(ϕ(A)ϕ(B))=0 \qquad(A,B\in\mathcal B(X)). \] Without assuming additivity, injectivity, or continuity, we show that there exists a function $γ: B(X)\mapsto \C \setminus\{0\}$ such that the map $ϕ$ takes one of the following forms: 1. there exists a bounded linear or conjugate-linear, invertible operator $S:X\to X$ such that : \label{eq:direct} ϕ(A)=γ(A)SAS^{-1},\qquad \text{ for all } \quad A\in B(X), 2. the space $X$ is reflexive and there exists a bounded linear or conjugate-linear, invertible operator $S:X^*\to X$ such that \label{eq:dual} ϕ(A)=γ(A)SA^*S^{-1},\qquad \text{ for all } \quad A\in B(X).