Maps Completely Preserving Involutions and Maps Completely Preserving Drazin Inverse

Let and be infinite dimensional Banach spaces over the real or complex field , and let and be standard operator algebras on and , respectively. In this paper, the structures of surjective maps from onto that completely preserve involutions in both directions and that completely preserve Drazin inverse in both direction are determined, respectively. From the structures of these maps, it is shown that involutions and Drazin inverse are invariants of isomorphism in complete preserver problems.