On the Stable Perturbation and Nashed's Condition for Generalized Inverses

Let T be a bounded linear operator from a Banach space to a Banach space with closed range and let (T) over bar = T + delta T: Nashed's condition is that oI thorn (T + delta TT+)T-1 maps the null space of T into the range of T. The stable perturbation means that the intersection of the range of (T) over bar and the null space of the generalized inverse of T is {0}. We show that the stable perturbation is the same as Nashed's condition in the sense of duality.