Maps preserving the local spectrum of Jordan product of matrices

Let Mn(C)Mn(C) be the algebra of all n×nn×n complex matrices, and fix a nonzero vector x0∈Cnx0∈Cn. We show that a map φ   from Mn(C)Mn(C) into itself satisfiesσφ(T)φ(S)+φ(S)φ(T)(x0) = σTS+ST(x0), (T, S∈Mn(C)),σφ(T)φ(S)+φ(S)φ(T)(x0) = σTS+ST(x0), (T, S∈Mn(C)), if and only if there exists an invertible matrix A∈Mn(C)A∈Mn(C) such that Ax0=x0Ax0=x0 and φ(T)=±ATA−1φ(T)=±ATA−1 for all T∈Mn(C)T∈Mn(C).