Characterizations of Lie Higher Derivations on J-Subspace Lattice Algebras

Let $\mathcal{L}$ be a $\mathcal{J}$-subspace lattice on a Banach space $X$ over the real or complex field $\mathbb{F}$ and $\mathrm{Alg}\mathcal{L}$ be the associated $\mathcal{J}$-subspace lattice algebras. In this paper, we characterize the structure of a family $\{L_n\}_{n=0}^{\infty}: \mathrm{Alg}\mathcal{L}\rightarrow \mathrm{Alg}\mathcal{L}$ of linear mappings satisfying the condition $$L_n([A, B])=\sum_{i+j=n}[L_i(A), L_j(B)]$$ for any $A, B\in\mathrm{Alg}\mathcal{L}$ with $AB = 0$. Moreover, the family $\{L_n\}_{n=0}^{\infty}: \mathrm{Alg}\mathcal{L}\rightarrow \mathrm{Alg}\mathcal{L}$ of linear mappings satisfying $L_n([A, B]_{\xi})=\sum_{i+j=n}[L_i(A), L_j(B)]_{\xi}$ for any $A, B\in\mathrm{Alg}\mathcal{L}$ with $AB = 0$ and $1\neq \xi\in \mathbb{F}$ is also considered in the current work.