Additive Maps Preserving the Truncation of Operators
Received:September 24, 2020  Revised:April 02, 2021
Key Word: truncation of operator   operator equation   additive map   preserver
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 Author Name Affiliation Address Jie Yao School of Mathematics and Statistics， Shaanxi Normal University 陕西师范大学数学与统计学院 Guoxing Ji Shaanxi Normal University 陕西师范大学数学与统计学院
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Abstract:
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 said to be the truncation of $B$ in $\mathcal B(\mathcal H)$ if $A=P_{A}BP_{A^*}$, where $P_{A}$ and $P_{A^*}$ denotes the projection onto the closure 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.
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