Peripheral Local Spectrum Preservers of Jordan Products of Operators

We completely describe surjective maps preserving the peripheral local spectrum of Jordan product of operators. Let x(0) is an element of X and y(0) is an element of Y be two nonzero vectors. We show that a map phi from B(X) onto B(Y) satisfies gamma(ST)+TS(x(0)) = gamma phi(S)phi(T)+phi(T)phi(s)(Y-0), (T, S is an element of B(X)), if and only if there exists a bijective bounded linear map from X into Y such that A(x(0)) = y(0) and either phi(T) = -ATA(-1) for all T is an element of B(X) or phi(T) = AT A(-1) for all T is an element of B(X).