Additive Preservers of Drazin Invertible Operators with Bounded Index

Let B(X) be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space X. Given an integer n >= 1, we show that an additive surjective map Phi on B(X) preserves Drazin invertible operators of index non-greater than n in both directions if and only if F is either of the form Phi(T) = alpha Lambda T Lambda(-1) or of the form Phi(T) = alpha BT* B-1 where a is a non-zero scalar, A : X -> X and B : X* -> X are two bounded invertible linear or conjugate linear operators.