Lie derivable maps on

Let X be a Banach space of dimension greater than 1. We prove that if a map δ : B ( X ) → B ( X ) satisfies δ ( [ A , B ] ) = [ δ ( A ) , B ] + [ A , δ ( B ) ] for any A , B ∈ B ( X ) , then δ = D + τ , where D is an additive derivation of B ( X ) and the map τ : B ( X ) → F I vanishes at commutators [ A , B ] .