Preservers of eigenvalue inclusion sets

For a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gershgorin region, the Brauer region in terms of Cassini ovals, and the Ostrowski region. Characterization is obtained for maps Φ on n × n matrices satisfying S(Φ(A) − Φ(B)) = S(A − B) for all matrices A and B. From these results, one can deduce the structure of additive or (real) linear maps satisfying S(A) = S(Φ(A)) for every matrix A.