Maps preserving some spectral domains of Jordan product of operators

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