Jordan product and analytic core preservers

Abstract: Let $\mathcal{B} (X)$ be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space $X$. For an operator $ T \in \mathcal{B} (X)$, $K(T)$ denotes as usual the analytic core of $T$. We determine the form of surjective maps $ \phi $ on $ \mathcal{B} (X)$ satisfying $$ K(\phi(T) \phi(S) + \phi (S) \phi (T)) = K(TS + ST) $$ for all $T, S \in \mathcal{B} (X)$.