Let A be a unital standard algebra on a complex Banach space X with dimX ≥ 2. The main result of this paper is to characterize the linear maps δ, τ : A → B(X) satisfying Aτ (B) + δ(A)B = 0 whenever A, B ∈ A are such that AB = 0. As application of our main result, we determine the linear map δ : A → B(H) that has one of the following properties for A, B ∈ A: if AB* = 0, then Aδ(B)* + δ(A)B* = 0, or if A*B = 0, then A*δ(B)+δ(A)*B = 0, where A is a unital standard operator algebras on a Hilbert space H such that A is closed under the adjoint operation. We also provide other applications of the main result.
