Surjective maps preserving local spectral radius

Abstract

Let B(X) be the algebra of all bounded linear operators on a complex Banach space X. Let x 0 is a nonzero fixed vector in X. We give the concrete form of every surjective map φ from B(X) into its self, such that the local spectral radius of φ(T )φ(S) + φ(R) at x 0 equals the local spectral radius of T S + R at x 0 . We do not assume φ to be linear, or even additive.