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

Let $\mathcal{L}$ be a subspace lattice on a Banach space $X$ and let $\delta :\mathrm{A}\mathrm{l}\mathrm{g}\mathcal{L}\to B(X)$ be a linear mapping. If $\vee \{L\in \mathcal{L}:L_{-}⊉L\}=X$ or $\wedge{L_-:L\in $\mathcal{L}$, L_-\nsupseteq 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{A}\mathrm{l}\mathrm{g}$$\mathcal{L}$)^\prime$\mathrm{.}If$\vee{L\in $\mathcal{L}$: L_-\nsupseteq L}=X$or$\wedge{L_-:L\in $\mathcal{L}$, L_-\nsupseteq L}=(0)$and$\delta$satisfies$\delta(AB+BA)=\delta(A)B+A\delta(B)+\delta(B)A+B\delta(A)$whenever$AB=0$,weobtainthat$\delta$isageneralizedderivationand$\delta(I)A\in($\mathrm{A}\mathrm{l}\mathrm{g}$$\mathcal{L}$)^\prime$forevery$A\in $\mathrm{A}\mathrm{l}\mathrm{g}$$\mathcal{L}$$\mathrm{.}Wealsoprovethatif$\vee{L\in $\mathcal{L}$: L_-\nsupseteq L}=X$and$\wedge{L_-:L\in $\mathcal{L}$, L_-\nsupseteq L}=(0)$, then $\delta$ is a local generalized derivation if and only if $\delta$ is a generalized derivation.