Multiplicative Lie triple derivations on standard operator algebras

Abstract

Abstract Let χ be a Banach space of dimension n > 1 and 𝔘 ⊂ 𝔅(χ) be a standard operator algebra. In the present paper it is shown that if a mapping d : 𝔘 → 𝔘 (not necessarily linear) satisfies d ( [ [ U , V ] , W ] ) = [ [ d ( U ) , V ] , W ] + [ [ U , d ( V ) , W ] ] + [ [ U , V ] , d ( W ) ] d\left( {\left[ {\left[ {U,V} \right],W} \right]} \right) = \left[ {\left[ {d\left( U \right),V} \right],W} \right] + \left[ {\left[ {U,d\left( V \right),W} \right]} \right] + \left[ {\left[ {U,V} \right],d\left( W \right)} \right] for all U, V, W ∈ 𝔘, then d =ψ + τ, where ψ is an additive derivation of 𝔘 and τ : 𝔘 → 𝔽I vanishes at second commutator [[U, V ], W ] for all U, V, W ∈ 𝔘. Moreover, if d is linear and satisfies the above relation, then there exists an operator S ∈ 𝔘 and a linear mapping τ from 𝔘 into 𝔽I satisfying τ ([[U, V ], W ]) = 0 for all U, V, W ∈ 𝔘, such that d(U) = SU − US + τ (U) for all U ∈ 𝔘.