Additive Lie (ξ-Lie) derivations and generalized Lie (ξ-Lie) derivations on nest algebras

For a scalar ξ, a notion of (generalized) ξ-Lie derivations is introduced which coincides with the notion of (generalized) Lie derivations if ξ=1. Some characterizations of additive (generalized) ξ-Lie derivations on the triangular algebras and the standard operator subalgebras of Banach space nest algebras are given. It is shown, under some suitable assumption, that an additive map L is an additive (generalized) Lie derivation if and only if it is the sum of an additive (generalized) derivation and an additive map from the algebra into its center vanishing all commutators; is an additive (generalized) ξ-Lie derivation with if and only if it is an additive (generalized) derivation satisfying L(ξA)=ξL(A) for all A.