Characterization of derivations on B(X) by local actions

Let A be a unital algebra and M be a unital A-bimodule. A linear map δ: A→M is said to be Jordan derivable at a nontrivial idempotent P∈A if δ(A)∘B+A∘δ(B)=δ(A∘B) for any A,B∈A with A ○ B= P, here A ○ B = AB + BA is the usual Jordan product. In this article, we show that if A=AlgN is a Hilbert space nest algebra and M=B(H), or A=M=B(X), then, a linear map δ:A→M is Jordan derivable at a nontrivial projection P∈N or an arbitrary but fixed nontrivial idempotent P∈B(X) if and only if it is a derivation. New equivalent characterization of derivations on these operator algebras was obtained.