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.
Categories: math.RA math.OA