Mappings that preserve pairs of operators with zero triple Jordan product

Let FF be a field and n⩾3n⩾3. Suppose S1,S2⊆Mn(F)S1,S2⊆Mn(F) contain all rank-one idempotents. The structure of surjections ϕ:S1→S2ϕ:S1→S2 satisfying ABA=0⇔ϕ(A)ϕ(B)ϕ(A)=0ABA=0⇔ϕ(A)ϕ(B)ϕ(A)=0 is determined. Similar results are also obtained for (a) subsets of bounded operators acting on a complex or real Banach space XX, (b) the space of Hermitian matrices acting on n  -dimensional vectors over a skew-field DD, (c) subsets of self-adjoint bounded linear operators acting on an infinite dimensional complex Hilbert space. It is then illustrated that the results can be applied to characterize mappings ϕ on matrices or operators such thatF(ABA)=F(ϕ(A)ϕ(B)ϕ(A))for allA,Bfor functions F such as the spectral norm, Schatten p-norm, numerical radius and numerical range, etc.