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 \(\int _I \Vert \dot{\gamma }(t)\Vert dt\).