Subspaces with or without a common complement

TL;DR

This paper investigates the geometric structure of pairs of subspaces in a Hilbert space, focusing on those with or without a common complement, and characterizes their properties using operator theory and Grassmannian geometry.

Contribution

It characterizes the topological and geometric structure of subspace pairs with or without common complements in Hilbert spaces, including their parametrization and manifold properties.

Findings

Delta is an open set with connected components parametrized by subspace dimensions.

Gamma is a smooth submanifold characterized by semi-Fredholm indices.

The connected component of Delta with infinite dimensions is dense in its space.

Abstract

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