Maps on matrices preserving local spectra

Abstract

For a square matrix T and a nonzero vector e ∈ ℂ n , let σ T (x) be the local spectrum of T at e. Characterization is obtained for surjective maps φ on ℳ n (ℂ) satisfying σφ(T)−φ(S)(e) ⊆ σ T−S (e) for all matrices T and S. The same description is obtained by supposing that σ T−S (e) ⊆ σφ(T)−φ(S)(e) for all matrices T and S, without the surjectivity assumption on φ. Continuous maps from ℳ n (ℂ) onto itself that preserve the local spectral radius distance at a nonzero fixed vector are also characterized.