Lie Higher Derivations on Nest Algebras

Abstract

Let $\mathcal{N}$ be a nest on a Banach space $X$, and Alg$\mathcal{N}$ be the associated nest algebra. It is shown that if there exists a non-trivial element in $\mathcal{N}$ which is complemented in $X$, then $D = (L_n)_{n∈N}$ is a Lie higher derivation of Alg$\mathcal{N}$ if and only if each $L_n$ has the form $L_n(A) = τ_n(A) + h_n(A)I$ for all $A ∈ {\rm Alg}\mathcal{N}$, where $(τ_n)_{n∈N}$ is a higher derivation and $(h_n)_{n∈N}$ is a sequence of additive functionals satisfying $h_n([A, B]) = 0$ for all $A, B ∈ {\rm Alg}\mathcal{N}$ and all $n ∈ \boldsymbol{N}$.