A note on geodesics of projections in the Calkin algebra

Abstract

Let ${\mathcal {C}}({\mathcal {H}})={\mathcal {B}}({\mathcal {H}})/{\mathcal {K}}({\mathcal {H}})$ be the Calkin algebra (${\mathcal {B}}({\mathcal {H}})$ the algebra of bounded operators on the Hilbert space ${\mathcal {H}}$, ${\mathcal {K}}({\mathcal {H}})$ the ideal of compact operators, and $\pi :{\mathcal {B}}({\mathcal {H}})\rightarrow {\mathcal {C}}({\mathcal {H}})$ the quotient map), and ${\mathcal {P}}_{{\mathcal {C}}({\mathcal {H}})}$ the differentiable manifold of selfadjoint projections in ${\mathcal {C}}({\mathcal {H}})$. A projection p in ${\mathcal {C}}({\mathcal {H}})$ can be lifted to a projection $P\in {\mathcal {B}}({\mathcal {H}})$: $\pi (P)=p$. We show that, given $p,q\in {\mathcal {P}}_{{\mathcal {C}}({\mathcal {H}})}$, there exists a minimal geodesic of ${\mathcal {P}}_{{\mathcal {C}}({\mathcal {H}})}$ which joins p and q if and only if there exist lifting projections P and Q such that either both $N(P-Q\pm 1)$ are finite dimensional, or both are infinite dimensional. The minimal geodesic is unique if $p+q- 1$ has trivial anhihilator. Here the assertion that a geodesic is minimal means that it is shorter than any other piecewise smooth curve $\gamma (t)\in {\mathcal {P}}_{{\mathcal {C}}({\mathcal {H}})}$, $t\in I$, joining the same endpoints, where the length of $\gamma$ is measured by $\int _I \Vert \dot{\gamma }(t)\Vert dt$.