Maps preserving the fixed points of products of operators

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

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