Linear maps preserving the set of Fredholm operators

Abstract

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. In this paper we characterize surjective linear maps Φ: FB(H)→ B(H) preserving the set of Fredholm operators in both directions. As an application we prove that Φ preserves the essential spectrum if and only if the ideal of all compact operators is invariant under Φ and the induced linear map φ on the Calkin algebra is either an automorphism, or an anti-automorphism. Moreover, we have, either ind (Φ)(T)) = ind(T) or ind(Φ (T)) = -ind(T) for every Fredholm operator T.