Lie triple and Jordan derivable mappings on nest algebras

Abstract

Let 𝒩 be a nontrivial nest on a Banach space X over the complex field ℂ, assume that there exists a nontrivial element in 𝒩 which is complemented in X. Let Alg 𝒩 be the associated nest algebra. In this article, we show that if δ is a Lie triple derivable mapping from Alg 𝒩 into B(X) then for any A, B ∈ Alg 𝒩 there exists a λ A,B (depending on A and B) in ℂ such that δ(A + B) = δ(A) + δ(B) + λ A,B I, and δ = D + τ, where D is an additive derivation from Alg 𝒩 into B(X) and τ is a mapping from Alg 𝒩 into ℂI such that τ(A + B) = τ(A) + τ(B) + λ A,B I and τ([[A, B], C]) = 0 for all A, B, C ∈ Alg 𝒩. We also show that if δ is a Jordan derivable mapping from Alg 𝒩 into B(X) then δ is an additive derivation.