Linear Maps on Standard Operator Algebras Characterized by Action on Zero Products

Abstract

Let \({\mathcal {A}}\) be a unital standard algebra on a complex Banach space \({\mathcal {X}}\) with dim\({\mathcal {X}}\ge 2\). The main result of this paper is to characterize the linear maps \(\delta , \tau : {\mathcal {A}}\rightarrow B({\mathcal {X}})\) satisfying \( A \tau ( B) + \delta ( A) B = 0\) whenever \(A,B\in {\mathcal {A}}\) are such that \(AB=0\). As application of our main result, we determine the linear map \(\delta : {\mathcal {A}}\rightarrow B({\mathcal {H}})\) that has one of the following properties for \(A,B\in {\mathcal {A}}\): if \(AB^{\star }=0\), then \(A\delta (B)^{\star }+\delta (A)B^{\star }=0\), or if \(A^{\star }B=0\), then \( A^{\star }\delta (B)+\delta (A)^{\star }B=0 \), where \({\mathcal {A}}\) is a unital standard operator algebras on a Hilbert space \({\mathcal {H}}\) such that \( {\mathcal {A}} \) is closed under the adjoint operation. We also provide other applications of the main result.