Maps preserving the local spectral subspace of Jordan product of operators

Abstract

Let X be a complex Banach space with dim X ≥ 4, B(X) the algebra of all bounded linear operators on X and λ_0 a fixed complex scalar. In the present paper, we characterize all maps ϕ on B(X) with range contains the operators of rank at most six that satisfying

            X_{ϕ(A)ϕ(T)+ϕ(T)ϕ(A)}({λ_0}) = X_{AT+TA(}{λ_0})

for all A, T ∈ B(X).