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Order preserving transformations of the Hilbert grassmannian

Mark Pankov
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Abstract

Let $H$ be a separable real Hilbert space. Denote by ${\mathcal G}_{\infty}(H)$ 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 ${\mathcal G}_{\infty}(H)$ can be extended to an automorphism of the lattice of closed subspaces of $H$. It follows from Mackey's result \cite{Mackey} that automorphisms of this lattice are induced by invertible bounded linear operators.

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Mark Pankov. 2006-05-14. Order preserving transformations of the Hilbert grassmannian. https://arxiv.org/abs/math/0605363

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