Additive Maps Preserving the Truncation of Operators
Received:September 24, 2020  Revised:April 27, 2021
Key Word: truncation of operator   operator equation   additive map   preserver
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11771261).
 Author Name Affiliation Jie YAO School of Mathematics and Statistics, Shaanxi Normal University, Shaanxi 710119, P. R. China Guoxing JI School of Mathematics and Statistics, Shaanxi Normal University, Shaanxi 710119, P. R. China
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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.