Lie-Type Derivations of Nest Algebras on Banach Spaces

Abstract

Let $\mathcal{X}$ be a Banach space over the complex field $\mathbb{C}$ and $\mathcal{B}(\mathcal{X})$ be the algebra of all bounded linear operators on $\mathcal{X}$. Let $\mathcal{N}$ be a non-trivial nest on $\mathcal{X}$, $\mathrm{Alg}\mathcal{N}$ be the nest algebra associated with $\mathcal{N}$, and $L:\mathrm{Alg}\mathcal{N}⟶\mathcal{B}(\mathcal{X})$ 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 $A_1,A_2,\cdots ,A_n\in \mathrm{Alg}\mathcal{N}$ if and only if there exist a linear derivation $D:\mathrm{Alg}\mathcal{N}⟶\mathcal{B}(\mathcal{X})$ and a linear mapping $H:\mathrm{Alg}\mathcal{N}⟶\mathbb{C}I$ vanishing on each $(n-1)$-th commutator $p_n(A_1,A_2,\cdots ,A_n)$ for all $A_1,A_2,\cdots ,A_n\in \mathrm{Alg}\mathcal{N}$ such that $L(A)=D(A)+H(A)$ for all $A\in \mathrm{Alg}\mathcal{N}$.

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