Changping Xiong | Searcharxiv

arXiv2014

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

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

math.OA↗

arXiv2013

Linear mappings of local preserving-majorization on matrix algebras

Let $\M_{n\times n}$ be the algebra of all $n\times n$ matrices. For $x,y\in {R}^{n}$ it is said that $x$ is majorized by $y$ if there is a double stochastic matrix $A\in {M}_{n\times n}$ such that $x=Ay$ (denoted by $x\prec y$). Suppose that $Φ$ is a linear mapping from ${R}^{n}$ into ${R}^{n}$, which is said to be strictly isotone if $Φ(x)\prec Φ(y)$ whenever $x\prec y$. We say that an element $α\in {R}^{n}$ is a strictly all-isotone point if every strictly isotone $φ$ at $α$ (i.e. $Φ(α)\precΦ(y)$ whenever $x\in {R}^{n}$ with $α\prec x$, and $Φ(x)\precΦ(α)$ whenever $x\in {R}^{n}$ with $x\prec α$) is a strictly isotone. In this paper we show that every $α=(α_{1},α_{2},...,α_{n})\in {R}^{n}$ with $α_{1}>α_{2}>...>α_{n}$ is a strictly all-isotone point.

math.QA↗