Orthogonal apartments in Hilbert Grassmannians

Let H be a complex Hilbert space. Denote by Gk(H) the Grassmannian consisting of k-dimensional subspaces of H. Every orthogonal apartment of Gk(H) is defined by a certain orthogonal base of H and consists of all k-dimensional subspaces spanned by subsets of this base. Orthogonal apartments can be characterized as maximal sets of mutually compatible elements of Gk(H). In the case when H is infinite-dimensional, we prove the following: if f is a bijective transformation of Gk(H) such that f and f−1 send orthogonal apartments to orthogonal apartments (in other words, f preserves the compatibility relation in both directions), then f is induced by an unitary or antiunitary operator on H. Suppose that dim⁡H=n is finite and not less than 3. For n≠2k (except the case when n=6 and k is equal to 2 or 4) we show that every bijective transformation of Gk(H) sending orthogonal apartments to orthogonal apartments is induced by an unitary or antiunitary operator on H. Our third result is the following: if n=2k≥8 and f is a bijective transformation of Gk(H) such that f and f−1 send orthogonal apartments to orthogonal apartments, then there is an unitary or antiunitary operator U such that for every X∈Gk(H) we have f(X)=U(X) or f(X) coincides with the orthogonal complement of U(X).