maps preserving the difference of minimum and surjectivity moduli of self-adjoint operators

Let h be a separable infinite dimensional complex hilbert space and sa(h) be the real jordan algebra of all bounded self-adjoint operators acting on h. in this paper, we study the general form of surjective non-linear maps ξ : sa(h) → sa(h), that preserve the difference of minimum and surjectivity moduli of self-adjoint operators in both directions. it turns out that ξ(p) = ep e∗ + r, (p, r ∈ sa(h)) where e : h → h, is either a bounded unitary or an anti-unitary operator