Linear Mappings Characterized by Action on Zero Products or Unit Products

Let A be a unital algebra andMbe a unital A-bimodule. We characterize the linear mappings δ and τ fromAintoM, satisfying δ(A)B+ Aτ(B) = 0 for every A, B ∈ A with AB = 0 when A contains a separating ideal T of M, which is in the algebra generated by all idempotents in A.We apply the result to P-subspace lattice algebras, completely distributive commutative subspace lattice algebras, and unital standard operator algebras. Furthermore, suppose that A is a unital Banach algebra andMis a unital Banach A-bimodule, we give a complete description of linear mappings δ and τ from A intoM, satisfying δ(A)B + Aτ(B) = 0 for every A, B ∈ A with AB = I .