论文标题
保存希尔伯特·格拉曼尼亚人的正交性
Orthogonality preserving transformations of Hilbert Grassmannians
论文作者
论文摘要
令$ h $为复杂的希尔伯特空间，让$ {\ Mathcal g} _ {k}（h）$为$ h $ $ k $二维子空间形成的Grassmannian。假设$ \ dim h> 2k $和$ f $是$ {\ Mathcal g} _ {k}（k}（h）$，即任何正交$ x，y \ in {\ Mathcal g} _ {k}（k k}（h）$ f（y f（y），如果$ \ dim h = n $是有限的，则$ n = mk+i $对于某些整数$ m \ ge 2 $和$ i \ in \ {0,1，\ dots，k-1 \} $（对于$ i = 0 $，我们有$ m \ m \ ge 3 $）。我们表明，如果$ i \ in \ in \ {0,1,1,2,3 \} $或$ m \ ge i+1 \ ge 5 $，则$ f $是由统一或反向自动运算符引起的两次培养。特别是，该语句以$ k \ in \ {1,2,3,4 \} $的价格保留，如果$ k \ ge 5 $，则恰好有$（k-4）（k-3）/2 $ n $的$ n $值，因此不满足上述条件。作为申请，我们将获得有关$ H $无限维度的情况的结果。
Let $H$ be a complex Hilbert space and let ${\mathcal G}_{k}(H)$ be the Grassmannian formed by $k$-dimensional subspaces of $H$. Suppose that $\dim H>2k$ and $f$ is an orthogonality preserving injective transformation of ${\mathcal G}_{k}(H)$, i.e. for any orthogonal $X,Y\in {\mathcal G}_{k}(H)$ the images $f(X),f(Y)$ are orthogonal. If $\dim H=n$ is finite, then $n=mk+i$ for some integers $m\ge 2$ and $i\in \{0,1,\dots,k-1\}$ (for $i=0$ we have $m\ge 3$). We show that $f$ is a bijection induced by a unitary or anti-unitary operator if $i\in \{0,1,2,3\}$ or $m\ge i+1\ge 5$; in particular, the statement holds for $k\in \{1,2,3,4\}$ and, if $k\ge 5$, then there are precisely $(k-4)(k-3)/2$ values of $n$ such that the above condition is not satisfied. As an application, we obtain a result concerning the case when $H$ is infinite-dimensional.
