Chow's theorem for Hilbert Grassmannians as a Wigner-type theorem

TL;DR

This paper characterizes transformations of infinite-dimensional Hilbert Grassmannians that preserve a specific adjacency relation, showing they are induced by unitary or anti-unitary operators or their compositions with orthocomplementation.

Contribution

It extends Wigner-type theorems to the setting of infinite-dimensional Grassmannians, describing all bijections preserving ortho-adjacency as induced by fundamental symmetries.

Findings

Transformations preserving ortho-adjacency are induced by unitary or anti-unitary operators.

Such transformations can also be composed with orthocomplementation.

Different components may correspond to different inducing operators.

Abstract

Let $H$ be an infinite-dimensional complex Hilbert space. Denote by ${\mathcal G}_{\infty}(H)$ the Grassmannian formed by closed subspaces of $H$ whose dimension and codimension both are infinite. We say that $X,Y\in {\mathcal G}_{\infty}(H)$ are {\it ortho-adjacent} if they are compatible and $X\cap Y$ is a hyperplane in both $X,Y$. A subset ${\mathcal C}\subset {\mathcal G}_{\infty}(H)$ is called an $A$-{\it component} if for any $X,Y\in {\mathcal C}$ the intersection $X\cap Y$ is of the same finite codimension in both $X,Y$ and ${\mathcal C}$ is maximal with respect to this property. Let $f$ be a bijective transformation of ${\mathcal G}_{\infty}(H)$ preserving the ortho-adjacency relation in both directions. We show that the restriction of $f$ to every $A$-component of ${\mathcal G}_{\infty}(H)$ is induced by a unitary or anti-unitary operator or it is the composition of the orthocomplementary map and a map induced by a unitary or anti-unitary operator. Note that the restrictions of $f$ to distinct components can be related to different operators.