Linear Maps Preserving the Set of Semi-Weyl 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 characterized the linear maps ϕ:B(H)→B(H), which are surjective up to compact operators preserving the set of left semi-Weyl operators in both directions. As an application, we proved that ϕ preserves the essential approximate point spectrum if and only if the ideal of all compact operators is invariant under ϕ and the induced map φ on the Calkin algebra is an automorphism. Moreover, we have ind(ϕ(T))=ind(T) if both ϕ(T) and T are Fredholm.