Pseudospectrum of ξξ‐Lie product on bounded linear operators

Let and be the algebra of all bounded linear operators on a complex Hilbert space and the Jordan algebra of all self‐adjoint operators in , respectively. In this paper, we give characterizations of rank one operators by the pseudospectrum on ‐Lie product of bounded linear operators and discuss some properties about the pseudospectrum. As applications, we obtain the structures of all surjective maps preserving the pseudospectrum of ‐Lie product on and , respectively.