Additive maps preserving Jordan zero-products on nest algebras

Let AlgN and AlgM be nest algebras associated with the nests N and M on Banach Spaces. Assume that N∈N and M∈M are complemented whenever N-=N and M-=M. Let Φ:AlgN→AlgM be a unital additive surjection. It is shown that Φ preserves Jordan zero-products in both directions, that is Φ(A)Φ(B)+Φ(B)Φ(A)=0⇔AB+BA=0, if and only if Φ is either a ring isomorphism or a ring anti-isomorphism. Particularly, all unital additive surjective maps between Hilbert space nest algebras which preserves Jordan zero-products are characterized completely.