Lie-Type Derivations of Nest Algebras on Banach Spaces

Abstract

Let $\mathcal{X}$ be a Banach space over the complex field $\mathbb{C}$ and $\mathcal{B(X)}$ be the algebra of all bounded linear operators on $\mathcal{X}$. Let $\mathcal{N}$ be a non-trivial nest on $\mathcal{X}$, ${\rm Alg}\mathcal{N}$ be the nest algebra associated with $\mathcal{N}$, and $L\colon {\rm Alg}\mathcal{N}\longrightarrow \mathcal{B(X)}$ be a linear mapping. Suppose that $p_n(x_1,x_2,\cdots,x_n)$ is an $(n-1)$-th commutator defined by $n$ indeterminates $x_1, x_2, \cdots, x_n$. It is shown that $L$ satisfies the rule $$ L(p_n(A_1, A_2, \cdots, A_n))=\sum_{k=1}^{n}p_n(A_1, \cdots, A_{k-1}, L(A_k), A_{k+1}, \cdots, A_n) $$ for all $A_1, A_2, \cdots, A_n\in {\rm Alg}\mathcal{N}$ if and only if there exist a linear derivation $D\colon {\rm Alg}\mathcal{N}\longrightarrow \mathcal{B(X)}$ and a linear mapping $H\colon {\rm Alg}\mathcal{N}\longrightarrow \mathbb{C}I$ vanishing on each $(n-1)$-th commutator $p_n(A_1,A_2,\cdots, A_n)$ for all $A_1, A_2, \cdots, A_n\in {\rm Alg}\mathcal{N}$ such that $L(A)=D(A)+H(A)$ for all $A\in {\rm Alg}\mathcal{N}$.