Lie triple and Jordan derivable mappings on nest algebras

Abstract

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. Keywords: nest algebrasLie triple derivable mappingsJordan derivable mappingsAMS Subject Classifications:: 47L3517B60 Acknowledgements The authors would like to thank the referee for his or her valuable comments and suggestions. The work is supported by National Natural Science Foundation of China (No. 11071188).