Journal of Operator Theory

Volume 79, Issue 1, Winter 2018 pp. 79-100.

Essentially orthogonal subspaces

Authors: Esteban Andruchow (1) and Gustavo Corach (2)
Author institution: (1) Instituto de Ciencias, Universidad Nacional de General Sarmiento, (1613) Los Polvorines and Instituto Argentino de Matematica Alberto P. Calderon, CONICET, (1083) Buenos Aires, Argentina
(2) Facultad de Ingenieria, Universidad de Buenos Aires, (1063) Buenos Aires and Instituto Argentino de Matematica Alberto P. Calderon, CONICET, (1083) Buenos Aires, Argentina

Summary: We study the set $\mathcal C$ consisting of pairs of orthogonal projections $P,Q$ acting in a Hilbert space $\mathcal H$ such that $PQ$ is a compact operator. These pairs have a rich geometric structure which we describe here. They are partitioned in three subclasses: $\mathcal C_0$ consists of pairs where $P$ or $Q$ have finite rank, $\mathcal C_1$ of pairs such that $Q$ lies in the restricted Grassmannian (also called Sato--Grassmannian) of the polarization $\mathcal H=N(P)\oplus R(P)$, and $\mathcal C_\infty$. We characterize the connected components of these classes: the components of $\mathcal C_0$ are parametrized by the rank, the components of $\mathcal C_1$ are parametrized by the Fredholm index of the pairs, and $\mathcal C_\infty$ is connected. We show that these subsets are (non-complemented) differentiable submanifolds of $\mathcal B(\mathcal H)\times \mathcal B(\mathcal H)$.

DOI: http://dx.doi.org/10.7900/jot.2016dec13.2138
Keywords: projections, pairs of projections, compact operators, Grasmann manifold

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