Preservers of the local spectral radius zero of Jordan product of operators

Abstract

Let B(X) be the algebra of all bounded linear operators on an infinite-dimensional complex Banach spaceX , and denote by rT (x) the local spectral radius of any operator T ∈ B(X) at any vector x ∈ X . In this paper, wecharacterize surjective maps φ on B(X) satisfyingrφ(T )φ(A)+φ(A)φ(T )(x) = 0 if and only if rT A+AT (x) = 0for all x ∈ X and A, T ∈ B(X).