Eduardo Chiumiento

Essentially commuting projections

Let $\mathcal{H}=\mathcal{H}_+\oplus\mathcal{H}_-$ be a fixed orthogonal decomposition of a Hilbert space, with both subspaces of infinite dimension, and let $E_+, E_-$ be the projections onto $\mathcal{H}_+$ and $\mathcal{H}_-$. We study the set $\mathcal{P}_{cc}$ of orthogonal projections $P$ in $\mathcal{H}$ which {\it essentially commute} with $E_+$, i.e.
$$
[P,E_+]=PE_+-E_+P \ \ \hbox{ is compact.}
$$
Using the projection $\pi$ onto the Calkin algebra, one sees that these projections $P\in\mathcal{P}_{cc}$ fall into nine classes.
Indeed, $\pi(P)$ can be represented as a $2 \times 2$ diagonal projection, where the diagonal elements can be $0$, $1$ or a proper projection. We define the {\it discrete classes}, when the diagonal elements are $0$ or $1$, which corresponds to finite rank projections, finite co-rank projections, the restricted Grassmannian of $\mathcal{H}_+$ and the restricted Grassmannian of $\mathcal{H}_-$. The connected components of each of these classes are parametrized by the integers. The five remaining classes are called {\it essential classes}, and they are connected. We show that the Hopf-Rinow Theorem holds in the discrete classes, but not in the essential classes. This is joint work with Esteban Andruchow and María Eugenia Di Iorio.

XXXVII Workshop on Geometric Methods in Physics	1-7.07.2018
VII School on Geometry and Physics	25-29.06.2018

Event sponsored by:
University of Bialystok