Let X and Y be two complex Banach spaces, and let B(X) denotes the algebra of all bounded linear operators on X. We characterize additive maps from B(X) onto B(Y ) compressing the pseudospectrum subsets Δ ϵ(.), where Δ ϵ(.) stands for any one of the spectral functions σ ϵ(.), σ l ϵ(.) and σ r ϵ (.) for some ϵ > 0. We also characterize the additive (resp. non-linear) maps from B(X) onto B(Y) preserving the pseudospectrum σ ϵ(.) of generalized products of operators for some ϵ > 0 (resp. for every ϵ > 0).