Orthogonality preserving transformations of Hilbert Grassmannians

Let H be a complex Hilbert space and let Gk(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 Gk(H), i.e. for any orthogonal X,Y∈Gk(H) the images f(X),f(Y) are orthogonal. Furthermore, 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+1)/2 values of n such that the above condition is not satisfied.