Published in Linear Algebra and Its Applications
Let X be a Banach space of dimension greater than 2. We prove that if δ:B(X)→B(X) is a linear map satisfyingδ([A,B])=[δ(A),B]+[A,δ(B)]for any A,B∈B(X) with AB=0 (resp. AB=P, where P is a fixed nontrivial idempotent), then δ=d+τ, where d is a derivation of B(X) and τ:B(X)→CI is a linear map vanishing at commutators [A,B] with AB=0 (resp. AB=P).