2024/03/16 by Mark Pankov, Pankov, Mark
Mathematics · #Advanced Algebra and Geometry #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2403.10886
openalex publication_date 2024/03/16 · openalex created_date 2024/03/21 · openalex updated_date 2026/07/28
Let H be an infinite-dimensional complex Hilbert space and let \mathcal G∞(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∞(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→ H such that for every X∈ \mathcal G∞(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∈ \mathcal G∞(H), then the same holds for all X∈ \mathcal G∞(H).