Maps on matrices compressing the local spectrum in the spectrum

Let Mn(C) be the algebra of all complex n×n matrices. Let x0 be a nonzero vector in Cn. We describe maps ϕ on Mn(C) (not linear or surjective) satisfyingσT±S(x0)⊆σ(ϕ(T)±ϕ(S)) for all T,S∈Mn(C). We also obtain a similar description by supposing that ϕ is surjective and σϕ(T)±ϕ(S)(x0)⊆σ(T±S) for all T,S∈Mn(C).