Additive Drazin inverse preservers (Q996300)

Let \(H\) be a real or complex Hilbert space and denote by \(B(H)\) the algebra of all bounded linear operators acting on \(H\). An element \(T\in B(H)\) is called Drazin invertible if there exists an element \(T^D\in B(H)\) and a positive integer \(k\) such that \[ TT^D=T^DT, \quad T^DTT^D=T^D, \quad T^{k+1}T^D=T^k. \] The operator \(T^D\) is unique and called the Drazin inverse of \(T\). The author characterizes the additive maps \(\phi:B(H)\to B(K)\) (\(H,K\) being infinite-dimensional real or complex Hilbert spaces) which preserve the Drazin inverse in the sense that \(\phi(T^D)=\phi(T)^D\) holds for every Drazin invertible operator \(T\in B(H)\). It is proved that if the range of \(\phi\) contains every rank-one idempotent in \(B(K)\) and \(\phi\) does not annihilate all rank-one idempotents in \(B(H)\), then \(\phi\) is of one of the forms \[ \phi(T)=\xi ATA^{-1}, \quad A\in B(H), \] \[ \phi(T)=\xi AT^{tr}A^{-1}, \quad A\in B(H), \] where \(\xi=\pm 1\) and \(A:H\to K\) is a bounded linear or conjugate-linear bijection. The finite-dimensional case is also considered.