Maps preserving the fixed points of products of operators

Abstract

Let $X$ be a complex Banach space with $\dim X\geq3$ and $B(X)$ the algebra of all bounded linear operators on $X$. Suppose $\phi:B(X)\longrightarrow B(X)$ is a surjective map satisfying the following property: $Fix(AB)=Fix(\phi(A)\phi(B)), (A, B\in B(X))$. Then the form of $\phi$ is characterized, where $Fix(T)$ is the set of all fixed points of an operator $T$.