Bilocal derivations of standard operator algebras

In this paper, we shall show the following two results: (1) Let $A$ be a standard operator algebra with $I$, if $\Phi$ is a linear mapping on $A$ which satisfies that $\Phi (T)$ maps $\ker T$ into $\operatorname {ran} T$ for all $T\in A$, then $\Phi$ is of the form $\Phi (T)=TA+BT$ for some $A,B$ in $B(X)$. (2) Let $X$ be a Hilbert space, if $\Phi$ is a norm-continuous linear mapping on $B(X)$ which satisfies that $\Phi (P)$ maps $\ker P$ into $\operatorname {ran} P$ for all self-adjoint projection $P$ in $B(X)$, then $\Phi$ is of the form $\Phi (T)=TA+BT$ for some $A,B$ in $B(X)$.