2023/02/02 by Mark Pankov, Pankov, Mark, Adam Tyc +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #FOS: Physical sciences #Mathematical Physics (math-ph) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2302.01077
openalex publication_date 2023/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let H be an infinite-dimensional complex Hilbert space. Denote by \mathcal G∞(H) the Grassmannian formed by closed subspaces of H whose dimension and codimension both are infinite. We say that X,Y∈ \mathcal G∞(H) are \it ortho-adjacent if they are compatible and X∩ Y is a hyperplane in both X,Y. A subset \mathcal C⊂ \mathcal G∞(H) is called an A-\it component if for any X,Y∈ \mathcal C the intersection X∩ 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∞(H) preserving the ortho-adjacency relation in both directions. We show that the restriction of f to every A-component of \mathcal G∞(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.