A note on geodesics of projections in the Calkin algebra

Let C(H) = B(H)/K(H) be the Calkin algebra (B(H) the algebra of bounded operators on the Hilbert space H, K(H) the ideal of compact operators, and pi : B(H) -> C(H) the quotient map), and PC( H) the differentiable manifold of selfadjoint projections in C(H). A projection p in C(H) can be lifted to a projection P is an element of B(H): pi(P) = p. We show that, given p, q is an element of P-C(H), there exists a minimal geodesic of P-C(H) which joins p and q if and only if there exist lifting projections P and Q such that either both N(P - Q +/- 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) is an element of P-C(H), t is an element of I, joining the same endpoints, where the length of gamma is measured by integral(I) parallel to(gamma) over dot(t)parallel to dt.