Lie-Type Derivations of Nest Algebras on Banach Spaces

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Let X be a Banach space over the complex field C and B(X) be the algebra of all bounded linear operators on X. Let N be a non-trivial nest on X, AlgN be the nest algebra associated with N, and L AlgN B(X) be a linear mapping. Suppose that pₙ(x₁,x₂,,xₙ) is an (n-1)-th commutator defined by n indeterminates x₁, x₂,, xₙ. It is shown that L satisfies the rule L(pₙ(A₁, A₂,, Aₙ))=∑ₖ₌₁ⁿpₙ(A₁,, Aₖ₋₁, L(Aₖ), Aₖ₊₁,, Aₙ) for all A₁, A₂,, Aₙ∈ AlgN if and only if there exist a linear derivation D AlgN B(X) and a linear mapping H AlgN CI vanishing on each (n-1)-th commutator pₙ(A₁,A₂,, Aₙ) for all A₁, A₂,, Aₙ∈ AlgN such that L(A)=D(A)+H(A) for all A∈ AlgN.