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