Abstract. In this paper, we shall show the following two results: (1) Let A be a standard operator algebra with I, if Φ is a linear mapping on A which satisfies that Φ(T ) maps ker T into ran T for all T ∈ A, then Φ is of the form Φ(T ) = T A + BT for some A, B in B(X). (2) Let X be a Hilbert space, if Φ is a norm-continuous linear mapping on B(X) which satisfies that Φ(P ) maps ker P into ran P for all self-adjoint projection P in B(X), then Φ is of the formIn what follows X stands for a Banach space (or Hilbert space) and X * for its norm dual. We denote by (x, f ) the duality pairing between elements f ∈ X * and x ∈ X, and we use the symbols "B(X)", "L(X)", "F (X)", "I" and "x ⊗ f" to denote the set of all linear bounded operators on X, the set of all linear mappings on X, the set of all finite rank operators on X, the identity operator and the rank one operator ( * , f)x on X, respectively.If A is a Banach algebra, and A 1 is a Banach subalgebra of A, we say that a linear mapping Φ : A 1 → A is a derivation if Φ(ab) = Φ(a)b + aΦ(b) for any a and b in A 1 . The derivation Φ is called inner if there exists an element a in A such that Φ(b) = ba − ab for any b in A 1 . We say that a linear mapping Φ : A 1 → A is a local derivation if for every a in A 1 , there exists a derivation δ a : A 1 → A, depending on a, such that Φ(a) = δ a (a). A linear mapping Φ is called a Jordan derivation if Φ(a 2 ) = aΦ(a) + aΦ(a) for every a in A 1 . We give the notion of bilocal derivation as follows: Definition 1. If A is a Banach subalgebra of B(X), a linear mapping Φ : A → B(X) is called a bilocal derivation if for every T in A and u in X, there exists a derivation δ T,u : A → B(X), depending on T and u, such that Φ(T )u = δ T,u (T )u.
