ADDITIVE MAPS PRESERVING ELEMENTS ANNIHILATED BY THE POLYNOMIALS XY Y X an‎d XY + Y X

Let H be a complex Hilbert space and B(H) denotes the algebra of all bounded linear operators on H. Suppose that : B(H) ! B(H) is an additive surjective map. We prove that if satises (A)(P) = (P)(A) if and only if AP = PA for 2 f1;1g and for all A and idempotent P in B(H), then is a *-automorphism