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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