Mappings on some reflexive algebras characterized by action on zero products or Jordan zero products

Abstract

Let $\mathcal{L}$ be a subspace lattice on a Banach space $X$ and let $\delta :\mathrm{Alg}\mathcal{L}\to B(X)$ be a linear mapping. If $\vee \{L\in \mathcal{L}:L_{-}⊉L\}=X$ or $\wedge \{L_{-}:L\in \mathcal{L},L_{-}⊉L\}=(0)$, we show that the following three conditions are equivalent: (1) $\delta (AB)=\delta (A)B+A\delta (B)$ whenever $AB=0$; (2) $\delta (AB+BA)=\delta (A)B+A\delta (B)+\delta (B)A+B\delta (A)$ whenever $AB+BA=0$; (3) $\delta$ is a generalized derivation and $\delta (I)\in (\mathrm{Alg}\mathcal{L}{)}^{\prime }$. If $\vee \{L\in \mathcal{L}:L_{-}⊉L\}=X$ or $\wedge \{L_{-}:L\in \mathcal{L},L_{-}⊉L\}=(0)$ and $\delta$ satisfies $\delta (AB+BA)=\delta (A)B+A\delta (B)+\delta (B)A+B\delta (A)$ whenever $AB=0$, we obtain that $\delta$ is a generalized derivation and $\delta (I)A\in (\mathrm{Alg}\mathcal{L}{)}^{\prime }$ for every $A\in \mathrm{Alg}\mathcal{L}$. We also prove that if $\vee \{L\in \mathcal{L}:L_{-}⊉L\}=X$ and $\wedge \{L_{-}:L\in \mathcal{L},L_{-}⊉L\}=(0)$, then $\delta$ is a local generalized derivation if and only if $\delta$ is a generalized derivation.

12 pages