(untitled) (Communications in Mathematical Research, 2021)
Let N be a nest on a Banach space X, and AlgN be the associated nest algebra. It is shown that if there exists a non-trivial element in N which is complemented in X, then D = (L_n)_n∈N is a Lie higher derivation of AlgN if and only if each L_n has the form L_n(A) = τ_n(A) + h_n(A)I for all A ∈ AlgN, 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 ∈ AlgN and all n ∈ N.