A note on geodesics of projections in the Calkin algebra

Abstract

Let ${\cal C}({\cal H})={\cal B}({\cal H}) / {\cal K}({\cal H})$ be the Calkin algebra (${\cal B}({\cal H})$ the algebra of bounded operators on the Hilbert space ${\cal H}$, ${\cal K}({\cal H})$ the ideal of compact operators and $\pi:{\cal B}({\cal H})\to {\cal C}({\cal H})$ the quotient map), and ${\cal P}_{{\cal C}({\cal H})}$ the differentiable manifold of selfadjoint projections in ${\cal C}({\cal H})$. A projection $p$ in ${\cal C}({\cal H})$ can be lifted to a projection $P\in{\cal B}({\cal H})$: $\pi(P)=p$. We show that given $p,q \in {\cal P}_{{\cal C}({\cal H})}$, there exists a minimal geodesic of ${\cal P}_{{\cal C}({\cal H})}$ which joins $p$ and $q$ if and only there exist lifting projections $P$ and $Q$ such that either both $N(P-Q\pm 1)$ are finite dimensional, or both 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 $\gamma(t) \in {\cal P}_{{\cal C}({\cal H})}$, $t \in I$, joining the same endpoints, where the length of $\gamma$ is measured by $\int_I \|\dot{\gamma}(t)\| d t$.