Additive Maps Preserving Local Spectrum

Let X be a complex Banach space, and let be the space of bounded operators on X. Given and x ∈ X, denote by σ_T (x) the local spectrum of T at x. We prove that if is an additive map such that then Φ (T)  =  T for all We also investigate several extensions of this result to the case of where The proof is based on elementary considerations in local spectral theory, together with the following local identity principle: given and x ∈X, if σ_S+R (x)  =  σ_T+R (x) for all rank one operators then S_x  =  T_x .