Let B(X) be the space of all bounded linear operators on complex Banach space X. For A ∈ B(X), we denote by F(A) the subspace of all fixed points of A. In this paper, we study and characterize all surjective maps φ on B(X) satisfying F(φ(T)φ(A) + φ(A)φ(T)) = F(T A + AT) for all A, T ∈ B(X).