ABSTRACT

Let H be a separable complex Hilbert space with dim H ≥ 3, ${\mathcal{B}}_s\left(H\right)$ be the Lie algebra of all bounded self-adjoint operators on H, and let $F:i{\mathcal{B}}_s\left(H\right)\to [d,\mathrm{\infty }]$ with $d\ge 0$ be a radial unitary similarity invariant function. In this paper, a structure feature is obtained for maps φ on ${\mathcal{B}}_s\left(H\right)$ satisfying $F\left(\phi \right(A\left)\phi \right(B)-\phi (B\left)\phi \right(A\left)\right)=F(AB-BA)$ for all $A,B\in {\mathcal{B}}_s\left(H\right).$ As applications, we show that, for a surjective map φ on ${\mathcal{B}}_s\left(H\right)$, the following conditions are equivalent: φ preserves the p-norm for some $1\le p<\mathrm{\infty }$ on Lie products; φ preserves the numerical radius on Lie products; φ preserves the pseudo-spectral radius on Lie products; there exists a unitary or conjugate unitary operator U on H, a sign function $h:{\mathcal{B}}_s\left(H\right)\to \{1,-1\}$ and a functional $g:{\mathcal{B}}_s\left(H\right)\to \mathbb{R}$ such that $\phi \left(T\right)=h\left(T\right)UT{U}^{\ast }+g\left(T\right)I$ for all $T\in {\mathcal{B}}_s\left(H\right)$. We also show that the following conditions are equivalent: φ preserves the numerical range on Lie products; φ preserves the pseudo spectrum on Lie products. Moreover, the concrete forms of the above preservers are given. The case $\dim H=2$ is also discussed.

摘要

设H是一个可分的复 Hilbert 空间，dim H ≥ 3，${\mathcal{乙}}_{秒}\left(H\right)$是H上所有有界自伴随算子的李代数，令$F：\mathrm{一世}{\mathcal{乙}}_{秒}\left(H\right)\to [d,\mathrm{\infty }]$ 和 $d\ge 0$是一个径向酉相似不变函数。在本文中，获得了映射φ上的结构特征${\mathcal{乙}}_{秒}\left(H\right)$ 满意 $F\left(\phi \right(\mathrm{一种}\left)\phi \right(乙)-\phi (乙\left)\phi \right(\mathrm{一种}\left)\right)=F(\mathrm{一种}乙-乙\mathrm{一种})$ 对所有人 $\mathrm{一种},乙\in {\mathcal{乙}}_{秒}\left(H\right).$作为应用，我们证明了，对于一个满射φ上${\mathcal{乙}}_{秒}\left(H\right)$，以下条件是等价的：φ为某些保留p范数$1\le 磷<\mathrm{\infty }$关于谎言产品；φ保留李乘积的数值半径；φ保留了 Lie 积的伪谱半径；在H上存在酉或共轭酉算符U，一个符号函数$H：{\mathcal{乙}}_{秒}\left(H\right)\to \{1,-1\}$ 和一个功能 $G：{\mathcal{乙}}_{秒}\left(H\right)\to \mathbb{电阻}$ 以至于 $\phi \left(吨\right)=H\left(吨\right)你吨{你}^{＊}+G\left(吨\right)\mathrm{一世}$ 对所有人 $吨\in {\mathcal{乙}}_{秒}\left(H\right)$. 我们还证明了以下条件是等价的：φ保留了李乘积的数值范围；φ保留了李乘积的伪谱。此外，还给出了上述保存器的具体形式。案子$\mathrm{暗淡}H=2$ 也在讨论中。