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 x0 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 )a tx0 equals the local spectral radius of TS + R at x0. We do not assume φ to be linear, or even additive.