Nonlinear maps preserving the minimum and surjectivity moduli

Let X and Y   be infinite-dimensional complex Banach spaces, and let B(X)B(X) (resp. B(Y)B(Y)) denote the algebra of all bounded linear operators on X (resp. on Y). We describe maps φ   from B(X)B(X) onto B(Y)B(Y) satisfyingc(φ(S)±φ(T))=c(S±T)c(φ(S)±φ(T))=c(S±T) for all S,T∈B(X)S,T∈B(X), where c(⋅)c(⋅) stands either for the minimum modulus, or the surjectivity modulus, or the maximum modulus. We also obtain analog results for the finite-dimensional case.