Linear Mappings Characterized by Action on Zero Products or Unit Products

Abstract

Let $\mathcal {A}$ be a unital algebra and $\mathcal {M}$ be a unital $\mathcal {A}$-bimodule. We characterize the linear mappings $\delta$ and $\tau$ from $\mathcal {A}$ into $\mathcal {M}$, satisfying $\delta (A)B+A\tau (B)=0$ for every $A,B \in \mathcal {A}$ with $AB=0$ when $\mathcal {A}$ contains a separating ideal $\mathcal {T}$ of $\mathcal {M}$, which is in the algebra generated by all idempotents in $\mathcal {A}$. We apply the result to $\mathcal {P}$-subspace lattice algebras, completely distributive commutative subspace lattice algebras, and unital standard operator algebras. Furthermore, suppose that $\mathcal {A}$ is a unital Banach algebra and $\mathcal {M}$ is a unital Banach $\mathcal {A}$-bimodule, we give a complete description of linear mappings $\delta$ and $\tau$ from $\mathcal A$ into $\mathcal M$, satisfying $\delta (A)B+A\tau (B)=0$ for every $A,B\in \mathcal {A}$ with $AB=I$.