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- W2576543452 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 $$ P_If=chi_If Q_Jf= left(chi_J hat{f}right)check{ } , fin L^2(mathbb{R}^n), $$ where $I, Jsubset mathbb{R}^n$ 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 ${cal B}({cal H})times {cal B}({cal H})$." @default.
- W2576543452 created "2017-01-26" @default.
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- W2576543452 date "2017-01-13" @default.
- W2576543452 modified "2023-09-27" @default.
- W2576543452 title "Essentially orthogonal subspaces" @default.
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