Pairs of Projections: Geodesics,Fredholm and Compact Pairs |
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Authors: | Esteban Andruchow |
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Institution: | 1. Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150, 1613?, Los Polvorines, Argentina 2. Instituto Argentino de Matemática, Saavedra 15, 3er. piso, 1083?, Buenos Aires, Argentina
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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 $$\begin{aligned} \dim (R(P)\cap N(Q))=\dim (R(Q)\cap N(P)). \end{aligned}$$ |
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