Orthogonal apartments in Hilbert Grassmannians. Finite-dimensional case

Abstract

Let $H$ be a complex Hilbert space of finite dimension $n\ge 3$. Denote by ${\mathcal G}_{k}(H)$ the Grassmannian consisting of $k$-dimensional subspaces of $H$. Every orthogonal apartment of ${\mathcal G}_{k}(H)$ is defined by a certain orthogonal base of $H$ and consists of all $k$-dimensional subspaces spanned by subsets of this base. For $n\ne 2k$ (except the case when $n=6$ and $k$ is equal to $2$ or $4$) we show that every bijective transformation of ${\mathcal G}_{k}(H)$ sending orthogonal apartments to orthogonal apartments is induced by an unitary or conjugate-unitary operator on $H$. The second result is the following: if $n=2k\ge 8$ and $f$ is a bijective transformation of ${\mathcal G}_{k}(H)$ such that $f$ and $f^{-1}$ send orthogonal apartments to orthogonal apartments then there is an unitary or conjugate-unitary operator $U$ such that for every $X\in {\mathcal G}_{k}(H)$ we have $f(X)=U(X)$ or $f(X)$ coincides with the orthogonal complement of $U(X)$.