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 $\pi :\mathcal{B}(\mathcal{H})\to \mathcal{C}(\mathcal{H})$ 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 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 $\gamma (t)\in {\mathcal{P}}_{\mathcal{C}(\mathcal{H})}$, $t\in I$, joining the same endpoints, where the length of $\gamma$ is measured by ${\int }_I\Vert \dot{\gamma }(t)\Vert dt$.