Maps preserving some spectral domains of Jordan product of operators

Abstract

Let X be an infinite-dimensional complex Banach space and let $\mathcal {B}(X)$ denote the algebra of all bounded linear operators on X. For an operator $T \in \mathcal {B}(X)$ the sets $\sigma _{1}(T), \sigma _{2}(T),$ and $\sigma _{3}(T)$ are called, respectively, the semi-Fredholm domain, the Fredholm domain, and the Weyl domain, of T in the spectrum, $\sigma (T)$. Given $i \in \{1,2,3\}$, the goal of this article is to describe the general form of all surjective maps $\phi$ on $\mathcal {B}(X)$ which satisfy

$$\begin{aligned} \sigma _{i}(\phi (A)\phi (T) +\phi (T)\phi (A)) = \sigma _{i}(AT + TA) \end{aligned}$$

for all $A, T \in \mathcal {B}(X)$.