Maps Preserving the Difference of Minimum and Surjectivity Moduli of Self-adjoint Operators

Let H be a separable infinite dimensional complex Hilbert space and SAH) 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 $ : SAH) → SAH), that preserve the difference of minimum and surjectivity moduli of self-adjoint operators in both directions. It turns out that É(P) = EPE* + R, (P, RESA(H)) +H, is either a bounded unitary or an anti-unitary oper where E: H ator.