Let be a real or complex Hilbert space and denote by the algebra of all bounded linear operators acting on . An element is called Drazin invertible if there exists an element and a positive integer such that
The operator is unique and called the Drazin inverse of . The author characterizes the additive maps ( being infinite-dimensional real or complex Hilbert spaces) which preserve the Drazin inverse in the sense that holds for every Drazin invertible operator . It is proved that if the range of contains every rank-one idempotent in and does not annihilate all rank-one idempotents in , then is of one of the forms
where and is a bounded linear or conjugate-linear bijection. The finite-dimensional case is also considered.