Characterizations of Lie Higher Derivations on J-Subspace Lattice Algebras

Abstract

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 $\{L_n{\}}_{n=0}^{\infty }:\mathrm{Alg}\mathcal{L}\to \mathrm{Alg}\mathcal{L}$ 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 $\{L_n{\}}_{n=0}^{\infty }:\mathrm{Alg}\mathcal{L}\to \mathrm{Alg}\mathcal{L}$ 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 $1\ne \xi \in \mathbb{F}$ is also considered in the current work.