论文标题
关节频谱在投影上收缩地图
Joint spectrum shrinking maps on projections
论文作者
论文摘要
令$ \ MATHCAL H $为有限的尺寸复杂的Hilbert Space，其尺寸$ n \ ge 3 $和$ \ MATHCAL P（\ MATHCAL H）$ $ \ MATHCAL H $上的一组预测。令$φ：\ MATHCAL P（\ MATHCAL H）\ to \ MATHCAL P（\ MATHCAL H）$为溢流地图。我们表明，$φ$缩小了当且仅当其为任何两个预测保存的关节频谱时，都会收缩任何两个预测的联合频谱，因此以特定方式在$ \ mathbb c $上的环形自动形态引起。此外，对于任意的$ k \ ge 3 $，$φ$缩小了任何$ k $投影的关节频谱，并且仅当它是由统一或反军事引起的。假设$ ϕ $是格拉曼（Grassmann）排名一号预测空间上的溢流地图。我们表明，$ ϕ $是保留任何$ n $排名一个预测的关节频谱，并且只有将其扩展到$ \ Mathcal p（\ Mathcal {h}）$上的冲销映射，这是任何两个项目的频谱。此外，对于任何$ k> n $，$ ϕ $都是关节频谱在任何$ k $等级的预测中都会缩小，并且只有当它是由单一或一个反军事引起的。
Let $\mathcal H$ be a finite dimensional complex Hilbert space with dimension $n \ge 3$ and $\mathcal P(\mathcal H)$ the set of projections on $\mathcal H$. Let $φ: \mathcal P(\mathcal H) \to \mathcal P(\mathcal H)$ be a surjective map. We show that $φ$ shrinks the joint spectrum of any two projections if and only if it is joint spectrum preserving for any two projections and thus is induced by a ring automorphism on $\mathbb C$ in a particular way. In addition, for an arbitrary $k \ge 3$, $φ$ shrinks the joint spectrum of any $k$ projections if and only if it is induced by a unitary or an anti-unitary. Assume that $ϕ$ is a surjective map on the Grassmann space of rank one projections. We show that $ϕ$ is joint spectrum preserving for any $n$ rank one projections if and only if it can be extended to a surjective map on $\mathcal P(\mathcal{H})$ which is spectrum preserving for any two projections. Moreover, for any $k >n$, $ϕ$ is joint spectrum shrinking for any $k$ rank one projections if and only if it is induced by a unitary or an anti-unitary.
