Maps preserving the fixed points of products of operators

TL;DR

This paper characterizes surjective maps on operator algebras that preserve the fixed point sets of operator products, revealing their specific algebraic form in complex Banach spaces of dimension at least three.

Contribution

It provides a complete description of maps preserving fixed point sets of products of operators on complex Banach spaces, a novel structural insight.

Findings

The map $$ is characterized explicitly.

Preservation of fixed point sets constrains the form of $$.

The result applies to complex Banach spaces with dimension ≥ 3.

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 $T$.