Essentially orthogonal subspaces

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: ${\cal C}_0$ which consists of pairs where $P$ or $Q$ have finite rank, ${\cal C}_1$ of pairs such that $Q$ lies in the restricted Grassmannian (also called Sato Grassmannian) of the polarization ${\cal H}=N(P)\oplus R(P)$, and ${\cal C}_\infty$. Belonging to this last subclass one has the pairs $$ P_If=\chi_If ,\ \ Q_Jf= \left(\chi_J \hat{f}\right)\check{\ } , \ \ f\in L^2(\mathbb{R}^n), $$ where $I, J\subset \mathbb{R}^n$ are sets of finite Lebesgue measure, $\chi_I, \chi_J$ denote the corresponding characteristic functions and $\hat{\ } , \check{\ }$ denote the Fourier-Plancherel transform $L^2(\mathbb{R}^2)\to L^2(\mathbb{R}^2)$ and its inverse. We characterize the connected components of these classes: the components of ${\cal C}_0$ are parametrized by the rank, the components of ${\cal C}_1$ are parametrized by the Fredholm index of the pairs, and ${\cal C}_\infty$ is connected. We show that these subsets are (non complemented) differentiable submanifolds of ${\cal B}({\cal H})\times {\cal B}({\cal H})$.