Characterizations of all-derivable points in $B(H)$

Abstract

Let ${\mathcal{K}}$ and ${\mathcal{H}}$ be two Hilbert space, and let $B({\mathcal{K}},{\mathcal{H}})$ be the algebra of all bounded linear operators from ${\mathcal{K}}$ into ${\mathcal{H}}$. We say that an element $G\in B({\mathcal{H}},{\mathcal{H}})$ is an all-derivable point in $B({\mathcal{H}},{\mathcal{H}})$ if every derivable linear mapping $\varphi$ at $G$ (i.e. $\varphi(ST)=\varphi(S)T+S\varphi(T)$ for any $S,T\in B(H)$ with $ST=G$) is a derivation. Let both $\varphi: B({\mathcal{H}},{\mathcal{K}})\rightarrow B({\mathcal{H}},{\mathcal{K}})$ and $\psi: B({\mathcal{K}},{\mathcal{H}})\rightarrow B({\mathcal{K}},{\mathcal{H}})$ be two linear mappings. In this paper, the following results will be proved : if $Y\varphi(W)=\psi(Y)W$ for any $Y\in B({\mathcal{K}},{\mathcal{H}})$ and $W\in B({\mathcal{H}},{\mathcal{K}})$, then $\varphi(W)=DW$ and $\psi(Y)=YD$ for some $D\in B({\mathcal{K}})$. As an important application, we will show that an operator $G$ is an all-derivable point in $B({\mathcal{H}},{\mathcal{H}})$ if and only if $G\neq 0$.