p-Schatten commutators of projections

Abstract

Let ${\mathcal {H}}={\mathcal {H}}_+\oplus {\mathcal {H}}_-$ be a fixed orthogonal decomposition of the complex separable Hilbert space ${\mathcal {H}}$ in two infinite-dimensional subspaces. We study the geometry of the set ${\mathcal {P}}^p$ of selfadjoint projections in the Banach algebra

$${\mathcal {A}}^p=\{A\in {\mathcal {B}}({\mathcal {H}}): [A,E_+]\in {\mathcal {B}}_p({\mathcal {H}})\},$$

where $E_+$ is the projection onto ${\mathcal {H}}_+$ and ${\mathcal {B}}_p({\mathcal {H}})$ is the Schatten ideal of p-summable operators ($1\le p <\infty$). The norm in ${\mathcal {A}}^p$ is defined in terms of the norms of the matrix entries of the operators given by the above decomposition. The space ${\mathcal {P}}^p$ is shown to be a differentiable $C^\infty$ submanifold of ${\mathcal {A}}^p$, and a homogeneous space of the group of unitary operators in ${\mathcal {A}}^p$. The connected components of ${\mathcal {P}}^p$ are characterized, by means of a partition of ${\mathcal {P}}^p$ in nine classes, four discrete classes, and five essential classes: (1) the first two corresponding to finite rank or co-rank, with the connected components parametrized by these ranks; (2) the next two discrete classes carrying a Fredholm index, which parametrizes their components; (3) the remaining essential classes, which are connected.