Isometries of Grassmann spaces, II
Isometries of Grassmann spaces, II
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DOI：
10.1016/j.aim.2018.05.012
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发表时间：
2018-05
影响因子：
1.7
通讯作者：
G. Geh'er;P. vSemrl
中科院分区：
文献类型：
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作者：
G. Geh'er;P. vSemrl
Botelho，Jamison和Molnár [1]，Gehér和Šemrl [4]最近描述了Hilbert空间H上固定有限秩的所有投影的Grassmann空间的满射等距的一般形式。作为一个直接的结果，我们可以刻画一个固定有限余秩的投影的格拉斯曼空间的满射等距。本文解决了当H是可分的时，无限秩和无限余秩的所有投影的集合P∞（H）上满射等距的剩余结构问题。证明技术与以前的完全不同，并且基于对格拉斯曼P∞（H）中测地线的研究。然而，同样的方法在有限秩投影的情况下给出了另一种证明。
Abstract Botelho, Jamison, and Molnár [1], and Gehér and Šemrl [4] have recently described the general form of surjective isometries of Grassmann spaces of all projections of a fixed finite rank on a Hilbert space H. As a straightforward consequence one can characterize surjective isometries of Grassmann spaces of projections of a fixed finite corank. In this paper we solve the remaining structural problem for surjective isometries on the set P∞(H) of all projections of infinite rank and infinite corank when H is separable. The proof technique is entirely different from the previous ones and is based on the study of geodesics in the Grassmannian P∞(H). However, the same method gives an alternative proof in the case of finite rank projections.