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

Abstract

Let H be an infinite-dimensional complex Hilbert space. Denote by G ∞ ( H ) the Grassmannian formed by closed subspaces of H whose dimension and codimension both are infinite. We say that X , Y ∈ G ∞ ( H ) are ortho-adjacent if they are compatible and X ∩ Y is a hyperplane in both X, Y. A subset C ⊂ G ∞ ( H ) is called an A-component if for any X , Y ∈ C the intersection X ∩ Y is of the same finite codimension in both X, Y and C is maximal with respect to this property. Let f be a bijective transformation of G ∞ ( H ) preserving the ortho-adjacency relation in both directions. We show that the restriction of f to every A-component 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 A-components can be related to different operators.