Additive local invertibility preservers

Let L(X) be the algebra of all bounded linear operators on a complex Banach space X, and for a nonzero vector x is an element of X and T is an element of L(X), let sigma(T)(x) denote the local spectrum of T at x. We characterize additive surjective maps phi on L(X) which satisfy 0 is an element of sigma(phi(T))(x) if and only if 0 is an element of sigma(phi(T))(x) for every x is an element of X and T is an element of L(X). Extensions of this result to the case of different Banach spaces are also established. As application, additive maps from L(X) onto itself that preserve the inner local spectral radius zero of operators are classified.