Maps preserving the idempotency of products of operators

Maps preserving the idempotency of products of operators

Let H and K be infinite dimensional Hilbert spaces, while B(H) and B(K) denote the algebras of all linear bounded operators on H and K , respectively. We characterize the forms of additive mappings from B(H) into B(K) that preserve the nonzero idempotency of Jordan products of operators in both directions.