秩2投影集的几何结构

The geometric structure of rank 2 projections in 4-dimensional Hilbert space

Wigner定理指出Hilbert空间中秩1投影集上的等距满射由酉算子或反酉算子导出,Geher与Semrl将Wigner定理推广到了秩k的情形.Geher与Semrl在证明过程中利用了秩k投影集的几何性质刻画投影的正交性.讨论了4维空间H上秩2投影集P_(2)(H)的几何结构,指出P_(2)(H)中两两距离为1的投影至多有6个.

The classical Wigner's theorem states that every surjective isometry on rank one projection set in a Hilbert space is induced by a unitary or an anti-unitary operator.The theorem is generalized to the case of rank n by Geher and Semrl.In the proof process,the geometric properties of rank k projections set are used to characterize the orthogonality of projections.The geometric structure of the rank 2 projections set on 4 dimensional space H is discussed.It is pointed out that there are at most 6 projections with a pairwise distance of 1 in P_(2)(H).