Orthogonal apartments in Hilbert Grassmannians

Let H be a complex Hilbert space. Denote by g(k) (H) the Grassmannian consisting of k-dimensional subspaces of H. Every orthogonal apartment of 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. Orthogonal apartments can be characterized as maximal sets of mutually compatible elements of g(k)(H). In the case when H is infinite-dimensional, we prove the following: if f is a bijective transformation of g(k)(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 g(k)(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 g(k)(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 is an element of g(k) (H) we have f (X) = U(X) or f (X) coincides with the orthogonal complement of U(X). (C) 2016 Elsevier Inc. All rights reserved.