Linear maps between operator algebras preserving certain spectral functions (Q380290)

Let \(B(H)\) be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space \(H\), \(K(H)\) the closed ideal of all compact operators on \(H\), and \(\phi\) a surjective linear map on \(B(H)\) with \(\phi(I)-I \in K(H)\). The authors proved that, if \(\phi\) preserves the set of upper semi-Weyl operators and the set of all normal eigenvalues in both directions, then it is an automorphism. The authors also consider the relation between linear maps preserving the set of upper semi-Weyl operators and linear maps preserving the set of left invertible operators.