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

Abstract

Let H be a separable infinite dimensional complex Hilbertspace and SA(H) be the real Jordan algebra of all bounded self-adjointoperators acting on H. In this paper, we study the general form ofsurjective non-linear maps : SA(H) ! SA(H), that preserve thedifference of minimum and surjectivity moduli of self-adjoint operatorsin both directions. It turns out that(P) = EPE+ R; (P;R 2 SA(H))where E : H ! H, is either a bounded unitary or an anti-unitary operator.