Surjective maps preserving quasi-nilpotence of products of operators

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 $\phi :\mathcal{B}(X)\to \mathcal{B}(X)$ satisfying $$r(AB)=0⟺r(\phi (A)\phi (B))=0(A,B\in \mathcal{B}(X))\mathrm{.}$$ Without assuming additivity, injectivity, or continuity, we show that there exists a function $\gamma :B(X)\mapsto \text{C}\setminus \{0\}$ such that the map $\phi$ 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 $S:X^{\ast }\to X$ such that \label{eq:dual} φ(A)=γ(A)SA^*S^{-1},\qquad $\text{ for all }$ \quad A\in B(X).