Preservers of local spectra of matrix sums

Let Mn(C) be the set of n×n complex matrices, and for a nonzero vector e∈Cn and T∈Mn(C), let σT(e) denote the local spectrum of T at e. We characterize surjective maps ϕ on Mn(C) which satisfy σϕ(T)+ϕ(S)(e)⊆σT+S(e) for all matrices T and S. We arrive at the same conclusion by supposing that σT+S(e)⊆σϕ(T)+ϕ(S)(e) for all matrices T and S, without the surjectivity assumption on ϕ. Continuous maps from Mn(C) onto itself that preserve the local spectral radius of the sum of matrices at a nonzero fixed vector are also characterized.