Essentially orthogonal subspaces

ANDRUCHOW, ESTEBAN; CORACH, GUSTAVO

JOURNAL OF OPERATOR THEORY

THETA FOUNDATION

Lugar: Bucharest; Año: 2018 vol. 79 p. 79 - 100

0379-4024

We study the set C consisting of pairs of orthogonal projectionsP,Q acting in a Hilbert space H such that PQ is a compact operator. Thesepairs have a rich geometric structure which we describe here. They are partitionedin 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 characterizethe connected components of these classes: the components of C0 areparametrized by the rank, the components of C1 are parametrized by the Fredholmindex of the pairs, and C¥ is connected. We show that these subsets are(non-complemented) differentiable submanifolds of B(H) B(H).