Orthogonality preserving transformations of Hilbert Grassmannians

2020/01/19 by Pankov, Mark
#FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2001.06883

Abstract

Let H be a complex Hilbert space and let \mathcal G_k(H) be the Grassmannian formed by k-dimensional subspaces of H. Suppose that dim H>2k and f is an orthogonality preserving injective transformation of \mathcal G_k(H), i.e. for any orthogonal X,Y∈ \mathcal G_k(H) the images f(X),f(Y) are orthogonal. If dim H=n is finite, then n=mk+i for some integers m≥ 2 and i∈ \0,1,…,k-1\ (for i=0 we have m≥ 3). We show that f is a bijection induced by a unitary or anti-unitary operator if i∈ \0,1,2,3\ or m≥ i+1≥ 5; in particular, the statement holds for k∈ \1,2,3,4\ and, if k≥ 5, then there are precisely (k-4)(k-3)/2 values of n such that the above condition is not satisfied. As an application, we obtain a result concerning the case when H is infinite-dimensional.

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