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

Let H be an infinite-dimensional complex Hilbert space. Denote by $ {\mathcal G}_{\infty }(H) $ G infinity(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) $ X,Y is an element of G infinity(H) are ortho-adjacent if they are compatible and $ X\cap Y $ X boolean AND Y is a hyperplane in both X, Y. A subset $ {\mathcal C}\subset {\mathcal G}_{\infty }(H) $ C subset of G infinity(H) is called an A-component if for any $ X,Y\in {\mathcal C} $ X,Y is an element of C the intersection $ X\cap Y $ X boolean AND Y is of the same finite codimension in both X, Y and $ {\mathcal C} $ C is maximal with respect to this property. Let f be a bijective transformation of $ {\mathcal G}_{\infty }(H) $ G infinity(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.