Additive Maps Preserving the Truncation of Operators

DOI：10.3770/j.issn:2095-2651.2022.01.008

 作者 单位 姚洁 陕西师范大学数学与统计学院, 陕西 西安 710119 吉国兴 陕西师范大学数学与统计学院, 陕西 西安 710119

设$\mathcal H$是复Hilbert空间, $\mathcal B(\mathcal H)$是$\mathcal H$上有界线性算子组成的代数. 设$A,B\in\mathcal B(\mathcal H)$. 若$A=P_ABP_{A^*}$, 则称$A$是$B$的截断, 其中$P_A$和$P_{A^*}$分别表示$A$和$A^*$的值域闭包上的正交投影. 本文我们给出了$\mathcal B(\mathcal H)$上双边保持算子截断的可加映射的构造.

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ the algebra of all bounded linear operators on $\mathcal{H}$. An operator $A$ is called the truncation of $B$ in $\mathcal B(\mathcal H)$ if $A=P_{A}BP_{A^*}$, where $P_{A}$ and $P_{A^*}$ denote projections onto the closures of $R(A)$ and $R(A^*)$, respectively. In this paper, we determine the structures of all additive surjective maps on $\mathcal{B}(\mathcal{H})$ preserving the truncation of operators in both directions.