Essentially orthogonal subspaces

2017/01/13 by Andruchow, Esteban, Corach, Gustavo
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1701.03737

Abstract

We study the set \cal C consisting of pairs of orthogonal projections P,Q acting in a Hilbert space \cal H such that PQ is a compact operator. These pairs have a rich geometric structure which we describe here. They are parted in three subclasses: \cal C₀ which consists of pairs where P or Q have finite rank, \cal C₁ of pairs such that Q lies in the restricted Grassmannian (also called Sato Grassmannian) of the polarization \cal H=N(P)⊕ R(P), and \cal C_∞. Belonging to this last subclass one has the pairs P_If=χ_If , Q_Jf= (χ_J f)\check , f∈ L²(ℝⁿ), where I, J⊂ ℝⁿ are sets of finite Lebesgue measure, χ_I, χ_J denote the corresponding characteristic functions and , \check denote the Fourier-Plancherel transform L²(ℝ²)→ L²(ℝ²) and its inverse. We characterize the connected components of these classes: the components of \cal C₀ are parametrized by the rank, the components of \cal C₁ are parametrized by the Fredholm index of the pairs, and \cal C_∞ is connected. We show that these subsets are (non complemented) differentiable submanifolds of \cal B(\cal H)× \cal B(\cal H).

Related

• Subspaces with or without a common complement
• Orthogonal apartments in Hilbert Grassmannians. Finite-dimensional case
• Orthogonal apartments in Hilbert Grassmannians
• Orthogonality preserving transformations of Hilbert Grassmannians
• The matched projection and geodesics of the Grassmann manifold