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