数学科学系

Department of Mathematical Sciences

Numerical range of Lie-product of operators and Lie numerical range preservers

报告题目:Numerical range of Lie-product of operators and Lie numerical range preservers

 

报告人:侯晋川教授 (太原理工大学)

 

时间:2014917日(星期三)10:30-11:30

 

地点:理科楼数学系A304

 

摘要:Denote by $W(A)$ the numerical range of a bounded linear operator $A$, and $[A,B] = AB-BA$ the Lie product of two operators $A$ and $B$. Let $H,K$ be complex Hilbert spaces of dimension $\geq 2$ and $\Phi:{\mathcal B}(H)\to{\mathcal B}(K)$ be a map whose range contains  all operators of rank $\leq 2$. It is shown that $\Phi$ satisfies that $ {W}([\Phi(A),\Phi(B)])= {W}([A,B])$ for any $A,B\in{\mathcal B}(H)$ if and only if $\dim H = \dim K$, there exist $\varepsilon \in \{1,-1\}$, a functional $h: \cB(H)\rightarrow \IC$, a unitary operator $U\in{\mathcal B}(H,K)$, and a set ${\mathcal S}$ of operators in ${\mathcal B}(H)$, that consists of operators of the form $aP + bI$ for an orthogonal projection $P$ on $H$ if the dimension of $H$ is at least 3, such that $$\Phi(A)=\begin{cases}\varepsilon UAU^*+h(A)I &  $if$ \ A \in{\mathcal B}(H)\setminus \cS,\cr-\varepsilon UAU^* + h(A)I & $if$ \ A \in \cS,\cr\end{cases}$$ or $$\Phi(A) = \begin{cases} i\varepsilon UA^tU^*+h(A)I &  $if$ \ A\in{\mathcal B}(H) \setminus \cS,\cr-i\varepsilon UA^tU^*+h(A)I &   $if$ \ A\in\cS, \cr \end{cases}$$ where $A^t$ is the transpose of $A$ with respect to an orthonormal basis of $H$. The proof of this result depends on the classifications of operators $A$ or operator pairs $A, B$ with some symmetric properties of $W([A,B])$ that are of independent interest.

 

报告人简介:侯晋川教授主要从事数学的教学和科研工作,研究方向为数学中的算子理论、算子代数,物理和信息科学中的量子计算及量子信息理论。主持承担过十多项国家级科研项目和国际合作项目,发表创新性学术论文200余篇;出版专著1部,译著2部,工具参考书多本。发表论文中有被SCI收录130余篇。 作为第一完成人,获得第二届中国青年科技奖,山西省科技进步一等奖二项; 山西省自然科学二等奖一项; 享受国务院特殊津贴, 被评为山西省优秀专家,山西省第二届科技功臣称号。获得的其他荣誉还有:全国做出突出贡献的回国留学人员、 山西省优秀回国留学人员奖、全国优秀教师、中央直接联系的高级专家等。

 

联系人:崔建莲