标题： 以子空间等价对形式的非满射Wigner型定理 显示英文标题

标题： A non-surjective Wigner-type theorem in terms of equivalent pairs of subspaces

摘要： 设$H$为一个无限维的复希尔伯特空间，设${\mathcal G}_{\infty}(H)$为$H$的所有闭子空间的集合，其维数和余维数均为无限。 我们研究${\mathcal G}_{\infty}(H)$的变换（不一定为满射），将每一对子空间映射为等价的一对子空间；若存在一个线性等距变换将其中一对子空间映射到另一对子空间，则这两对子空间是等价的。 设$f$为这样的变换。 我们证明存在一个唯一的至多相差一个标量的线性或共轭线性等距同构$L:H\to H$，使得对于每个$X\in {\mathcal G}_{\infty}(H)$，像$f(X)$是$L(X)$与某个与$L$的值域正交的闭子空间$O(X)$的和。 当$H$可分时，我们给出以下充分条件以断言$f$是由线性或共轭线性等距同构引起的：如果对于某个$X\in {\mathcal G}_{\infty}(H)$，$O(X)=0$成立，那么对于所有$X\in {\mathcal G}_{\infty}(H)$同样成立。 显示英文摘要

摘要： Let $H$ be an infinite-dimensional complex Hilbert space and let ${\mathcal G}_{\infty}(H)$ be the set of all closed subspaces of $H$ whose dimension and codimension both are infinite. We investigate (not necessarily surjective) transformations of ${\mathcal G}_{\infty}(H)$ sending every pair of subspaces to an equivalent pair of subspaces; two pairs of subspaces are equivalent if there is a linear isometry sending one of these pairs to the other. Let $f$ be such a transformation. We show that there is a unique up to a scalar multiple linear or conjugate linear isometry $L:H\to H$ such that for every $X\in {\mathcal G}_{\infty}(H)$ the image $f(X)$ is the sum of $L(X)$ and a certain closed subspace $O(X)$ orthogonal to the range of $L$. In the case when $H$ is separable, we give the following sufficient condition to assert that $f$ is induced by a linear or conjugate linear isometry: if $O(X)=0$ for a certain $X\in {\mathcal G}_{\infty}(H)$, then the same holds for all $X\in {\mathcal G}_{\infty}(H)$.