Maps preserving the fixed points of products of operators

Abstract

Let $X$ be a complex Banach space with $\dim X\ge 3$ and $B(X)$ the algebra of all bounded linear operators on $X$. Suppose $\varphi :B(X)⟶B(X)$ is a surjective map satisfying the following property: $Fix(AB)=Fix(\varphi (A)\varphi (B)),(A,B\in B(X))$. Then the form of $\varphi$ is characterized, where $Fix(T)$ is the set of all fixed points of an operator $T$.

arXiv comment7 pages