Maps Completely Preserving Involutions and Maps Completely Preserving Drazin Inverse

Abstract

Let X and Y be infinite dimensional Banach spaces over the real or complex field 𝔽, and let 𝒜 and ℬ be standard operator algebras on X and Y, respectively. In this paper, the structures of surjective maps from 𝒜 onto ℬ 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.