Pairs of Projections: Geodesics, Fredholm and Compact Pairs

Abstract

A pair $(P, Q)$ of orthogonal projections in a Hilbert space $\mathcal{H}$ is called a Fredholm pair if

$$\begin{aligned} QP : R(P) \rightarrow R(Q) \end{aligned}$$

is a Fredholm operator. Let $\mathcal{F}$ be the set of all Fredholm pairs. A pair is called compact if $P-Q$ is compact. Let $\mathcal{C}$ be the set of all compact pairs. Clearly $\mathcal{C} \subset \mathcal{F}$ properly. In this paper it is shown that both sets are differentiable manifolds, whose connected components are parametrized by the Fredholm index. In the process, pairs $P, Q$ that can be joined by a geodesic (or equivalently, a minimal geodesic) of the Grassmannian of $\mathcal{H}$ are characterized: this happens if and only if

$$\begin{aligned} \dim (R(P)\cap N(Q))=\dim (R(Q)\cap N(P)). \end{aligned}$$