p-Schatten commutators of projections

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