Additive maps preserving local spectrum

Let X be a complex Banach space, and let L(X) be the space of bounded operators on X. Given T is an element of L(X) and x is an element of X, denote by sigma(T)(x) the local spectrum of T at x. We prove that if Phi : L(X) -> L(X) is an additive map such that sigma(Phi(T)) (X) = sigma(T(X)) (T is an element of L(X), x is an element of X), then Phi(T) = T for all T is an element of L(X). We also investigate several extensions of this result to the case of Phi : L(X) -> L(Y), where X not equal Y. The proof is based on elementary considerations in local spectral theory, together with the following local identity principle: given S, T is an element of L(X) and X is an element of X, if sigma(S+R)(X) = sigma(T+R)(X) for all rank one operators R is an element of L(X), then Sx = Tx.