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 $\phi$ at $G$ (i.e. $\phi (ST)=\phi (S)T+S\phi (T)$ for any $S,T\in B(H)$ with $ST=G$) is a derivation. Let both $\phi :B(\mathcal{H},\mathcal{K})\to B(\mathcal{H},\mathcal{K})$ and $\psi :B(\mathcal{K},\mathcal{H})\to B(\mathcal{K},\mathcal{H})$ be two linear mappings. In this paper, the following results will be proved : if $Y\phi (W)=\psi (Y)W$ for any $Y\in B(\mathcal{K},\mathcal{H})$ and $W\in B(\mathcal{H},\mathcal{K})$, then $\phi (W)=DW$ 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 $G\ne 0$.