Order preserving transformations of the Hilbert grassmannian

2006/05/14 by Mark Pankov, Pankov, Mark
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #math.CO #math.FA

paper · pdf · doi:10.48550/arxiv.math/0605363

arxiv created 2006/05/14 · arxiv updated 2009/12/01

Abstract

Let H be a separable real Hilbert space. Denote by \mathcal G_∞(H) the Grassmannian consisting of closed subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We show that every order preserving transformation of \mathcal G_∞(H) can be extended to an automorphism of the lattice of closed subspaces of H. It follows from Mackey's result \citeMackey that automorphisms of this lattice are induced by invertible bounded linear operators.

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