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 C consisting of pairs of
orthogonal projections P,Q acting in a Hilbert space 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: C0
consists of pairs where P or Q have finite rank, C1 of pairs
such that Q lies in the restricted Grassmannian (also called Sato--Grassmannian)
of the polarization H=N(P)⊕R(P), and
C∞.
We characterize the connected components of these classes: the components of
C0 are parametrized by the rank, the components of C1
are parametrized by the Fredholm index of the pairs, and C∞
is connected. We show that these subsets are (non-complemented)
differentiable submanifolds of B(H)×B(H).
DOI: http://dx.doi.org/10.7900/jot.2016dec13.2138
Keywords: projections, pairs of projections, compact operators, Grasmann manifold
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