保持秩1幂等元凸组合数值域的满射

Surjective Maps Preserving the Numerical Range of Convex Combinations of Rank-1 Idempotents

摘要: 设$\mathscr{H}$ 为可分复Hilbert空间，${ℐ}_1\left(\mathscr{H}\right)$ 表示$\mathscr{H}$ 上所有秩1幂等元的集合。映射$\psi$ 保持任意两个秩1投影算子的特定凸组合的数值半径当且仅当$\psi$ 保持任意两个秩1投影算子的转移概率。当$\mathscr{H}$ 维数为2时，设$\varphi :{ℐ}_1\left(\mathscr{H}\right)\to {ℐ}_1\left(\mathscr{H}\right)$ 为满射，且存在${\lambda }_0\in \left(0,1\right)$ 使得对任意$E,F\in {ℐ}_1\left(\mathscr{H}\right)$ 有$W\left({\lambda }_{0}E+\left(1-{\lambda }_0\right)F\right)=W\left({\lambda }_0\varphi \left(E\right)+\left(1-{\lambda }_0\right)\varphi \left(F\right)\right)$ 。通过证明$\varphi$ 保持秩1投影算子集，可得$\varphi$ 由某一酉算子或反酉算子诱导。

Abstract: Let $\mathscr{H}$ be a separable complex Hilbert space, and ${ℐ}_1\left(\mathscr{H}\right)$ denote the set of all rank-1 idempotents on $\mathscr{H}$ . A map $\psi$ preserves the numerical radius of a specific convex combination of any two rank-1 projection operators if and only if $\psi$ preserves the transition probability of any two rank-1 projection . Assume $\dim \mathscr{H}=2$ , let $\varphi :{ℐ}_1\left(\mathscr{H}\right)\to {ℐ}_1\left(\mathscr{H}\right)$ be a surjective map, and suppose there exists ${\lambda }_0\in \left(0,1\right)$ such that $W\left({\lambda }_{0}E+\left(1-{\lambda }_0\right)F\right)=W\left({\lambda }_0\varphi \left(E\right)+\left(1-{\lambda }_0\right)\varphi \left(F\right)\right)$ for any $E,F\in {ℐ}_1\left(\mathscr{H}\right)$ . By proving that $\varphi$ preserves the set of rank-1 projection operators, it is concluded that $\varphi$ is induced by a unitary or anti-unitary operator.