Additive Drazin inverse preservers

Let H   be a real or complex Hilbert space and B(H)B(H) denote the Banach algebra of all bounded linear operators on H  . For T∈B(H)T∈B(H), if there exists an operator TD∈B(H)TD∈B(H) and a positive integer k such thatTTD=TDT,TDTTD=TD,Tk+1TD=Tk,then T   is said to be Drazin invertible, and TDTD is a Drazin inverse of T  . We say a map Φ:B(H)→B(H)Φ:B(H)→B(H) preserves Drazin inverse if Φ(TD)=Φ(T)DΦ(TD)=Φ(T)D for every Drazin invertible operator T∈B(H)T∈B(H). In this paper, we determine the structures of additive maps on B(H)B(H) preserving Drazin inverses.