# 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

(2) Facultad de Ingenieria, Universidad de Buenos Aires, (1063) Buenos Aires

**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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