Linear Maps Preserving Generalized Invertibility

Abstract

Let H be an infinite-dimensional complex separable Hilbert space and \( \mathcal{B}(H) \) the algebra of all bounded linear operators on H. Let \( \phi :\mathcal{B}(H) \to \mathcal{B}(H) \) be a bijective continuous unital linear map preserving generalized invertibility in both directions. Then the ideal of all compact operators is invariant under ϕ and the induced linear map on the Calkin algebra is either an automorphism or an antiautomorphism.