Let H be a separable complex Hilbert space. Denote by Gr(H) the Grassmann manifold of H. We study the following sets of pairs of elements in Gr(H):
D={(S,T) in Gr(H) x Gr(H): there exists Z in Gr(H) such that S+Z=T+Z=H (direct sum)},
which are pairs of subspaces that have a common complement, and
G=Gr(H) x Gr(H) - D
which are pairs of subspaces that do not admit a common complement.
We identify S ≈ P_S$, the subspace S with the orthogonal projection P_S onto S. Thus we may regard D and G as subsets of B(H) x B(H) (here B(H) denotes the algebra of bounded linear operators in H). We show that D is open, and its connected components are parametrized by the dimension and codimension of the subspaces. The connected component of D having both infinite dimensional and co-dimensional subspaces is dense in the corresponding component of G(H) x G(H), and contractible. On the other hand, G is a (closed) C^\infty submanifold of B(H) x B(H), and we characterize the connected components of G in terms of dimensions and semi-Fredholm indices.
Several examples of pairs in D and the connected components of G are given in Hilbert spaces of functions.